Hi,
So, one thing to consider, is don't have the speakers facing each other if possible. If they are going to mostly be covered, that will help a ton despite being "weather resistant" builds on outdoor speakers. Don't expect great quality from an outdoor speaker, there are good ones, but you pay a lot for average speakers due to the materials and there's compromise on that in audio quality of course. It won't sound bad, it just won't hold a candle to a well treated room with some good speakers properly set up. Most outdoor speakers in the 5.25 to 6.5" driver area have small cabinets, and are sealed, so don't expect bass (especially farther away from the speakers like opposite the pool). There are outdoor subs, they are down firing usually (and some are actually buried, if that's you're thing, not mine as that's a lot of work and digging it up to trouble shoot lol, no thanks). A basic receiver or multi-channel amp in the 50 watt+ area should handle these speakers for that area if you just want ambient audio but not like reference level sound (75db, or 85db depending on content goal). Hitting 65~70db should be pretty easy which is what you've described (not overpowering, not the only thing you hear).
Check out OSD outdoor speakers (6.5" drivers) and outdoor subwoofers (10" at least, maybe look at having two of them). Well within your budget. Pretty good builds. Subs add a huge amount of improvement over essentially outdoor bookshelves.
Any receiver will do the job on this basically. You can also just buy an integreated or poweramp and a DAC interface for the bluetooth if you want more control on higher power (probably not necessary). Whatever you choose you'll want to keep it fairly covered or it will become dust/bug city and eventually fail to humidity, etc. But if it's in a cabinet or well covered (but ventilated) it should last a good long time. I operate equipment in Florida outdoors for years like this, just covered from the elements and the bugs/humidity do their thing but I get years out of the equipment, so its not that bad.
Very best,