It looks like Brave on iOS does not support authentication through client SSL certificates like it does in the desktop app.
This is supported in Safari on iOS through configuration profiles: I can install a .p12 client certificate on my phone, and this will automatically be used on websites that have SSL client verification enabled.
After some searching I found that Apple does not expose these configuration profiles to other apps, but it should be possible to import client certificates into the Brave app keystore: https://developer.apple.com/library/archive/qa/qa1745/_index.html