Battleship/Cruiser missiles are fine. Use them against hull, like they're intended to. Use pulses and other turrets on shields.
Missiles were specifically changed from anti fighter to anti cap because you simply pressed Q lots and the enemy bombers/fighters died. Now, you require solaris turrets, good aiming with secondaries and/or flak to deal with enemy bombers. Not to mention, having escorts really boosts your survivability as a capital ship.