I peeked at the failed AJAX calls and I saw that the site was denying my traffic because I was classified as a high risk endpoint. The name of their WAF/filtering vendor was in the error responses or headers, so I found a free way to query that company's threat database for my home IP. There it was, my IP was marked high risk because it was associated with a tor relay. It's a silly threat correlation, in my opinion, but it was enough of an annoyance that I stopped running the relay so I didn't have to worry about flaky firewalls.
The ironic thing was, I could have probably fired up Tor Browser and tried new exit nodes until I found a fresh one their system didn't know about, just to finish updating my claim! I actually just tethered my laptop to my phone, I think.
PS I forgot to ask: does anyone know if Snowflake is as detectable as a normal bridge or relay node? If so, there's the risk that your home connection gets a risk/reputation strike against like mine did.