Does being in shield power mode affect your bombs' damage?

No the power settings only affect the damage of your primary lasers. Changing power settings has no effect on Railgun's, Missile's, Mine's or Drone's damage.

Drako already answered your question, but we can go further...

Switching to shield power gives you a greater shield regen and a greater shield max, while reducing your damage dealt with blasters, the rate at which your blaster pool regenerates, your maximum speed, and the rate at which your engine regenerates.

Numbers are here:

http://www.swtor.com/community/showthread.php?p=7296379

To put it another way:

For a beginner bomber almost all your damage will come from sources other than your primary weapons, so the effect is negligible.

For an advanced bomber pilot, secondary and systems damage will still make the majority of your damage, but against competent opponents your ability to actually kill with that damage is very low and being effective is much easier if you help things along with your primary weapons. In this case good power management can make a noticeable difference.

The basic principles of power management are really the same for all of the ships in GSF, and learning to do it well benefits you no matter which ship you choose to fly.

Unlike other ships though, bombers aren't seriously crippling themselves if they set up near cover and then just keep power to shields for the rest of the match.