The password reset stuff comes from Illuminate\Auth\Passwords\PasswordResetServiceProvider, Illuminate\Auth\Passwords\DatabaseTokenRepository, Illuminate\Auth\Passwords\TokenRepositoryInterface, and Illuminate\Auth\Passwords\PasswordBrokerManager (maybe more).
To make big changes to how password resets are handled, you might need to extend these and override some methods.
Just changing the DB connection for password resets shouldn't be that much work though...
PasswordBrokerManager is instantiating DatabaseTokenRepository and passing in the DB connection as seen here
It's using app['db']->connection(); Which is going to be the current connection...
Huh...
It might be that if your user model has the $connection specified that would work... alternatively, changing the default connection in PasswordController...
I'm out of time to investigate further, but I'm posting because I think this might be useful for you to debug further?