Cut it off? Aren't these things normally velcro? I'm just curious how this is not the case.
Most anyone around this forum will tell you to take it off, yes. As you identified there really is not use in any of the lifts you mentioned. The typical purpose it serves is to make the bar more comfortable for squatting. However, the proper position (I have the books, not the DVD, but I assume it's in there) is more important than a pad. A little chalk, and you're set.
Will you regret cutting the pad off? I don't think you'll want one, but if you are really concerned, here's some peace of minid: you can probably order a velcro one that will fit just fine.