It can easily be argued that in the stock game (whether patched or not), the military budget needed to wage war was too low - warring was very cost-effective. Current BetterAI OTOH seems to have gone too far in the other direction: it's easy to spiral down the unit spam path, leading to economic stagnation.
The ways to avoid this depend on the neighbourhood. You can go for two different gambles:
1) Go light on military and work diplomacy to make sure you still can keep out of wars. You need enough land to keep economic advantage, and might try to capitalize on that advantage by beelining an advanced military tech that allows you a window during which you can go on offensive with fewer units, units that are technologically superior to those of your neighbours.
2) Go all-out war. Mass units, throw the kitchen sink at your neighbour. You still need to make sure nobody is going to attack you while you're going for someone - possibly by making sure everyone is busy warring.
Neither of the above is a surefire strategy. They depend on the types of neighbours you have (warmongers or peaceful builders) and can still fail. If you do succeed, you should be on the way towards victory, but should you fail you can go reroll. Due to the high risk/high reward nature of the strategies and the early phase of the game they need to be executed, I find them distasteful. The game is decided too early, the rest either being slow stagnation to death or a cruise for victory. There's no balance, no competitive endgame.
Note that going up or down difficulty levels is not really a viable solution: either the gamble is not necessary (going down enough you can compete without either gamble) and the gamble is only a way to win early on (you can gamble without fear of losing, as loss only means you need to play competitive game to the end); or the gamble is necessary, victory not being possible without gambling (again not good - you have to try a desperate strategy with high risk of failure to have a chance at competitive endgame).
If for any reason you get caught in the unit spam spiral of death, you no longer have a choice: you have to get something out of the military budget, so you have to go throw the kitchen sink at a neighbour. Should you lose that gamble, you can reroll. Victorious war may lead to competitive endgame or cruise to victory, depending on what the other civs were doing. Again, this is not balanced.
Also remember that if there's more than one continent and the different continents end up on different paths (one going the military route, the other going peaceful route), severe imbalance will already follow due to civs on one continent suffering from some degree of economic stagnation over what the other continent is having. You can see a glimmer of this already in stock game when one continent has peaceful builders / techers that happily trade and speed up the tech tree with the other continent consisting of more aggressive leaders that are grumpy tech traders and rather build units and go to war. In stock game, it's nothing a human player can't overcome: conquer the aggressors, take hold of at least a major part of the continent, develop the land and race the techtree powered by more land than anyone else has. But if the military budgets get higher, the difference between the strategies gets sharper as well.
Unfortunatelly whatever the point of balance, the human player will find a way to exploit the AI. If the AI doesn't build enough units, the human player will simply roll over it. If the AI builds enough units to make this impossible, then the military expenses (both hammers and maintenace costs) will drag the economy down causing stagnation - most likely exploitable by working lighter allowing the AI civs to stagnate. Neither option allows for balanced game. Therefore I still conclude that the balance will be found in the economic advantages and disavantages the human player will have from building a military that will invariably be enough to conquer the AI civ.
As I said earlier, I don't have a solution that would make warring hard but possible - I don't know how to make the AI smart in warring. I'm not even sure if fighting smart is possible using the civ combat engine - it is quite limited afterall. I do believe that between humans, diplomacy would play a huge role with the wars that are started by deals being decided more by the dealings before than on the field using units - the units are just an endgame for the diplomacy. And there's no way for AI to match this, to participate in this level of diplomacy. Due to the diplomatic castration, units end up playing the role of diplomats as well as the role of armies. Which leads us back to problem #1: how to use them in smart way?
I can't say I'd envy Blake's or Iustus' position here. There are good reasons why the AI needs to mass units (so the human player won't take them out easily), but OTOH that leads to other problems (unit spam spiral of death). If a solution that satisfies all parties exists and is found, great. If not, then the division of AI behaviour using the Aggressive AI setting is a good way to go.