My understanding of the FLP result is that it only applies to algorithms without timeouts.

Because it restates that, without timeouts, in theory, you could be waiting forever for a message to arrive. It's formally proved and whatnot, but it's not a super-surprising result when you restate it in common terms.

For message delivery, it seems that FLP says that you need to give up at-least-once, because at-least-once ostensibly requires an infinite number of retries. I'm not sure how it would apply to at-most-once.

Exactly-once is, so far as I can tell, a bit of a strawman. But it's a strawman that we all fall for, the first time we break stuff into different systems.

As a disclaimer, I am still wrapping my head around this stuff. Don't rely on me. Not even if your name is Salvatore.