Get yourself a capacitor, that should help. If it doesn't I would suggest purchasing a Optima battery. About hurting the sub, it could eventually hurt the sub but it's going to take a toll on your electrical system.
*waits for everybody who thinks the big 3 will solve world hunger*
It's ntohing huge.You amp is simply drawing more current than your alternator can produce.You are getting less power to your sub from the voltage dip.