Yes, damaged battleships can bombard.
A single attacking warship (or any unit with an attack value of 1+) in a sea zone with just enemy transports will destroy all the transports. The undefended transports rule is there so you don’t have to keep rolling the dice to finish a battle that has only one possible outcome - the destruction of all the transports.
You can keep hitting them, but they can’t hit you back. It might take you ten additional rounds of combat, but eventually you will roll the number of hits needed. No point rolling the dice, just remove the transports.