Should I have a different crew for bombers than for fighters?
Quite right, on aircraft without gunners the number of experienced gunners along with their accuracy and precision are completely worthless.
It is more efficient to have a number of crew slots dedicated to bomber crews as these skills are expensive to upgrade and would be wasted on fighters.
Bear in mind though that some heavy fighters still have one or two gunners (the Beaufighter, for example) so plan accordingly. Spending some skill points on these gunners can turn the tide on pursuers.