For example, I’ve noticed that some websites start throwing captchas at me or even just straight-up refuse to load with 403: unauthorized errors because I have my router set up to load-balance across two Internet connections. (At least, that’s my guess as to why it’s happening.)

I maintain several multi-wan commercial setups and they don’t have this problem. I obviously don’t know what your setup is but I’d guess something is wrong with how its handling flows / connections. Once a connection is established between your edge and an internet resource that flow should remain “stuck” to whatever wan port it started with and it sounds like that isn’t happening.