Your OP is almost a year old, but I'm seeing the same thing right now in Brave. If I use Edge, when I try to add an in-stock item to my cart, up pops the captcha. In Brave, all I get is a quick circle and then nothing. Did you ever figure this out?
EDIT 1: If I try in a Brave Private Window, it works. So, it's got to be a setting or extension. I'll have to dig through them again.
EDIT 2: It looks like it was my DNS provider. For some reason, in Brave it said Custom. I set it to Cloudflare and then told it to use my current provider and now it works. Odd.