OP here - I get that the betas are to make sure a pregnancy is "sticking," but wouldn't it be better for our sanity just to get the first beta to confirm the pregnancy and then wait it out until the ultrasound.
That's fine! You can do that.

I had three betas and didn't feel my sanity slipping at any point. Some people who don't even do IVF/IUI have intense anxiety between the time they get a positive pee stick test and the time they go to the OB. There's no one right way to feel about this.
Your RE is not going to go out and track you down and force you to submit to betas. If you don't want to do them, just don't go back after that first test. Done.