New post for situation analysis, i think we have to stop with Genghis and take care of Hatshy (he should be pleased for the moment again too).
1.5k gold for Lib also looks very tempting, full value i think? If he gives all.
Thanks for playing
Okay, so things didn't go exactly as we hoped, but no biggie. It may play into our hands.
First, I agree we need to break off the Genghis war and deal with Hattie. She will have a peace treaty with Sury and Genghis for another 7-ish turns, so realistically she is only going to be plotting on us. We can't have a two front war with two beasts. I looked at the save, and Genghis will pay 175g for peace, so he's hurting bad (probably because we lost no units and he lost a city). With peace he may be pleased with us. If we raze more cities he won't be. He has taken Nongoma and will probably park a bunch of units there to try to retain it (lots of cultural pressure I think). Genghis and Brennus have OB, so he should be able to get back and fro. Later on, we can smack into that stack, and hopefully kill a good portion of his army, with another stack pouncing through Brennus' land.
Hattie has a bunch of damaged units in Seattle. We even have units right outside it. Some damaged, but with

left so they can heal this turn. The muskets and suchlike will probably move into the city, but only one musket has city guardian. Just about everybody else are damaged and/or won't get defensive bonuses (which are mostly gone anyway as the city is in revolt). It's on flatland.
Even our damaged units are mostly okay and can fight. We have at least 6 full-health cannons that can attack the city next turn. I think we can quite safely wipe her stack and raze the city
Units that don't need healing can smash some stuff for gold. Not sure how you can see the gold for Lib trade with Genghis, but it's certainly tempting to get all that stuff. We're losing around 100g/t, so at some point we will need more. Soon we'll have Rifling too, and can upgrade to Cavs. Since there are a good amount of melee units around, I suppose it's a good idea to keep a contingent of Conqs around? Even if we want to upgrade 20 of the current 25 Conqs, at 80g a pop that's only 1600g, which we can easily afford with our gold + Genghis' (we'd have about 3000g).
We then have a 10t guaranteed peace with Genghis, and can deal with the east. Hattie has no off-shore islands from what I can see, so even if she won't cap (perhaps because Cathy refuses to break free), we can wipe her out entirely. Then I would think Cathy would cap. Maybe success from the previous war counts too? Not sure how that stuff works. In any case she should be very weak. Then Brennus is over there (plus the city in the west), also without offshore islands. He won't be long for this world, even with Cuirs.
Genghis will get Physics in 3 turns. If he is at peace, maybe he is more likely to go for more peaceful techs like Economics, Communism or Biology instead of beelining Rifling? Even with Rifling we should be able to deal with him with CR3 (+C1?) cannons, although combat in the open wouldn't be pretty.
It can even be positive that Genghis is still strong, as that will make it easier to get the others below him in power level, and thereby easier cap them (although I have to assume it's still dang hard in an OCC conquest game - I've never done/played it before).
Once we've dealt with the east, we roll back west with our veteran army, plus the army pumping out of Madrid every single turn.
Think we'll become happy too once the war ends? Though may have left-over war weariness the instant we re-declare on Hattie. However, perhaps we can get some happies from Genghis? Don't recall what trades we had with him before. But if he becomes pleased post-war, it should be no problem to re-establish OB and some resources trades.
Sury will still be friendly and doesn't look as dangerous as Genghis, so hopefully we can deal with him okay after the east is secured and Genghis is done for.
I'm still optimistic about our chances, as we're essentially tech leaders where it matters.
(Edit: We can also get +2

by trading Corn from Brennus for 6 gpt. Also, Genghis won't have copper now, we don't need it, so we should be able to get something good out of that, like Sugar and Incense, unless he has traded away too much of it already.)