We'll eventually want to do this properly, but the current implementation includes a dummy CAPTCHA @meskio has created here, meaning no UX changes should be necessary in the short-term: tpo/anti-censorship/rdsys#182 (closed)
A simple UX change you could do is to remove the captcha window and keep everything as it is now. So users click on 'Request a bridge' and then the bridge gets configured. I guess it might be nice to add some confirmation banner/window to indicate it worked well.
Together with this change TB should call the settings API instead of the captcha moat one. Let me know if the settings API needs any changes for that to happen.
This issue is being covered by this other one, which's focused on Desktop and is currently waiting for review. After its validation, the flow will be mirrored for Android in Figma as well.
Another update: Android's version of the flow currently being designed in Figma.