There is no best, they all work well, though some synergies appeal to some people, and some strategies, more than others. Balseraph are really flexible and have a lot of different strengths they can play to. I describe them in order of worst to best though, based on my own opinion, though on any water map (or one with lakes) I would rank Octopus Overlords as the best, depending on how much you can use tsunamis when attacking cities.
Fellowship of the Leaves I think is often the weakest for them, unless you capture some elven workers/slaves. Only workers with the elven race can build tile improvements without destroying forests, and ancient forest upgrades are usually the main reason for adopting this civ. This works for a Balseraph builder game, large dominating cities. Without being able to build the right kind of workers directly, this can be tedious, but that doesn't mean it isn't without merit. The religious civic, Guardian of Nature, gives happiness for forests, and thus bigger cities. But even without it running, you can usually get quite a bit of happiness in many other ways, so the biggest advantage is the ancient forestry upgrades.
A better solution sometimes is to consider making lots of lumbermills first wherever possible (no elves needed), and then switching to Fellowship of the Leaves temporarily, until they have all become ancient forests. You can't build the lumbermills after they've changed, but existing ones remain and boost the ancient forest's production. They change automatically over (a short) time when it is your state religion. Once all or most have formed, the religion can be abandoned. While this is overkill builder syndrome in most games, if I'm playing a heavy forest map like arborea, I often do this at some point because I've got forests all over anyway.
Also, if you have a Priest of Leaves (with or without the religion), you can spam tigers when you're bored, not just for the culture bonus in cities, but throw them in the arena, maybe a few of them will turn out ok.
The next weakest in my opinion is Runes of Kilmorph, though they have some merit too. Using their civic, you can crank out excellent production in cities with bonuses to mine production. Worth considering on a highlands map if the mining bonus sounds appealing. Their biggest appeal might be earthquake, cast by tier 3 priests, which unfortifies all units in a tile, reducing the enemy's defensive bonuses. Combine that with shadow mana (to remove city defensive bonuses) and maelstrom (air 2, to soften up the defenders with damage), and you don't need to be slowed down by catapults when moving your armies (when casting haste which doesn't speed up catapult movement). Also, you can build Mines of Gal-Dur, giving you three free iron and you don't need to research the iron working tech to use them. However, if you're doing it because you want iron badly, you can convert to any other religion after building it, and still receive the benefit.
One major downside in my opinion of all the good civs is just that I like the Undercouncil way way better than I like the Overcouncil. That may just be personal, but to me it's a factor.
Empyrean gives you access to one of the most devastating heros in the game, and I like the mounted units too (they cast blind, which makes units unable to move for two turns). The only other main reason is if you are facing a lot of shadows (I only ever care about this against players really), then this will help neutralize them.
Order is a favorite of mine in general, I love the bless bonus from tier 2 priests early on, and Unyielding Order from the tier 3's is great, especially when conquering new cities. I don't think there's a strong synergy with Balseraphs though.
So then my favorites are Octopus Overlords, Council of Esus, and Ashen Veil. If there's a decent amount of water on the map, Octopus Overlords can be very powerful for anyone. Their most powerful features I think are Tsunami and water walking, which you don't need to keep the religion for - you can stockpile cultists and stygian guards and then convert to something else. Building lunatics relatively early in the game can hold you over too, so it's not bad as an early, intermediary religion. I usually skip lunatics and spam cultists though. Saves a tech too.
The other, maybe bigger, reason for Octopus Overlords is to build Hemah, the hero caster. Twincast is an arcane promotion only purchasable by heros, and doubles their summons. Instead of one fire elemental, they summon two, for instance. Puppets summoned with twincast get it too. So Hemah can summon two puppets per turn, which can in turn summon two creatures each, for four total summons per turn. If you choose Keelyn as a leader, they stick around an extra turn, and you can end up with godly numbers of elementals running around, it's sick. 8 summons per turn, 2 turns duration, you can maintain 16 by the third turn I think. All from one hero. It's dangerous stuff.
Another choice is Council of Esus, which gives you Gibbon Goetia as an arcane hero to play with puppets too. If you found it, you can get Nox Noctis the holy building which makes all your units invisible within your borders, except the ones in your city (don't need to follow the religion for this). Council of Esus as a religion plays totally differently, and can be very fun, up to you. It lets you sneak attack (if you have open borders and have moved your units in, they don't get kicked out when you declare war).
The downside is that nightshades are pretty weak for what they're used for, and anyone can build shadows regardless of religion. If you spam recon units, some of them will be created with the religion, and will be able to get the hidden nationality promotion, so rangers, beastmasters, and assassins can be used without declaring war. Also, you don't get any good priests out of it, but if you're never going to see incense, I guess it doesn't matter much. It's a pretty limited religion, with some special features. I do think it's the best religion for Amurites though, because they can learn spells. Without spells, I think they get owned under hidden nationality too easily. The surprise attack is probably one of the really nice strategic advantages though, plus you can use the hidden nationality units to soften up defenses or improvements first.
Ashen Veil is a pretty awesome religion in general. Tier 2 priests spamming ae fire, Sacrifice the Weak civic, powerful late game units. If you don't have horses, Rosier is a good mid-level mounted hero (there's one for Order too).
That said, Balseraph has so many great things going for it, it's well-played with any religion. There's so much variety that if priests are an afterthought, or a support unit, they're still going to do very well. For Balseraph specifically, Ashen Veil and Octopus Overlords have the best synergy when using Harlequins to taunt. They both have attacks that will damage every unit in the stack severely, which you can use to soften them up before taunting. Then with taunt, you can hopefully kill off most of the defenders without ever even attacking with your own units. Maelstrom and to an extent Wither can also accomplish this.
With the summoner trait (Keelyn), arcane units are very powerful. You don't need a religion to make use of that, but it does give greater weight to Gibbon and Hemah. If you really like flesh golems, octopus overlords summon the strongest strength sacrifices (krakens).
Many times I don't worry much about religions when playing Baseraphs, and just focus on switching religions a few times to get good support priests for my units (Order for Bless, Kilmorph for the strength increase, Overlords/Veil for ae damage support, Empyrean for mounted blinding units). I may go for a religion just so I can found it and get the holy city, both for the gold boost it brings when it spreads, and for the extra mana. What religion I end up with might depend more on how the game develops, than on a pre-determined plan.
Also, one of the most powerful Balseraph tactics has absolutely nothing to do with religion anyway (imo). Dominate is the tier 3 mind spell that permanently gives you one of your enemy's units, but if it's resisted, the caster has to buy the promotion again (they lose the mind 3 promotion). But puppets are disposable, so if all you do is run around mind controlling the enemy without risking losing the promotion on your archmages, there ends up being no one left to fight! Give me mages and harlequins for air and taunt, some nice defensive units to make swiss cheese of attacking defenders, and a few other magely goodies, and I'm happy. Mimics are sweet fun too.
So, in a nutshell, if you want militant offensive priests, go ashen veil or octopus overlords. You can do it temporarily for the tier 2's, or enjoy other perks of the religion (such as water walking (EDIT: I mean kraken submarines) or hell hounds) at tier 3.
If you want to maintain special civic advantages, ashen veil lets you build heavy specialist cities, kilmorph gives massive production from mines, fellowship of leaves gives happiness from forests and gives you ancient forests, order lets you control happiness with military units (like hereditary rule from bts) as well as unyielding order from tier 3 priests (no unhappiness in the city, like the Globe Theater in bts). Council of Esus lets you mount surprise attacks from within enemy borders, a la the Trojan Horse.
Don't let that steer you away from say, Empyrean though...if you don't rely much on a specific religion anyway, then a hero like Chalid might be all the reason you really need in picking a religion, and letting him lead your armies. The rest can sometimes be gravy.