If you are at war with a race, taking its worlds, and have it in a losing position, why would you have to spend influence to accept a peace treaty ? Should a peace treaty cost a variable amount of influence. Especially when they are calling YOU up to talk to you? Shouldn't you at that point GET influence (they were cool and let us live after we declared war on them).
Computer players seem to be able to see your exact military might in terms of # of hulls and firepower, WITHOUT an embassy. And refitting a ship counts as it not existing. So when/if I refit ships I can have a chain of wars declared because my ship firepower level dropped below a certain level.
Makes little sense if you have been a trustworthy trading partner.
Since this can put you below the 100 influence needed to declare peace, and since it is possible to have everyone in contact with you at war (like I said, they will ALL do it if they perceive your fleet being too weak) you can get locked out of diplomacy entirely.

We obviously have a lot more logic to add to the AI diplomatic interactions, which is one of the things we're working on right now.

I think that you should be able to use influence to "buy" things (like peace treaties) that the AI doesn't necessarily want, but I agree that such things shouldn't necessarily have a mandatory influence cost, especially if the AI wants it more than you do.

Influence might also be able to be used as a counter to a declaration of war/nullification of all treaties. Ie last minute diplomatic talks...

Influence should in a way be almost a tradable currency, an alien race wants something, and I do not, they sweeten the deal by offering influence...