I suspect that this isn't a big issue for them considering how small the userbase is that will be going through the steps you did to sign using your own keys.
My setup is a little unusual with the shenanigans to allow multiple clients to access a single Bridge instance, but "run Bridge on your Windows desktop and point Thunderbird at it" is a pretty well-advertised use-case in their own docs/advertisement... and Thunderbird has PGP built-in, to the point where it's actually got its own key management stuff and everything.
> Thunderbird has PGP built-in, to the point where it's actually got its own key management stuff and everything
This postdates Proton Bridge, btw. But yes, probably we could do more to play nicely with it. But, when Bridge was created, using both Thunderbird and Proton to manage your keys was indeed not an expected or intended use case.