It is illegal, and all it will take for you to have to quit is to bowl against someone who objects to it, because you'll have to stop.
If you bowl with a bunch of folks who don't care, you can get away with it. You're not going to have to forfeit your score or anything, but you can be made to quit.
The issue is the "defacing the approach" language of the rule includes you putting a substance on the approach. If you're having to re-powder as often as you say, that means the stuff is coming off your shoe and going somewhere, right? Ergo, the rule.
I would follow Strider's advice and do something different with your footwork. Two easiest things to do are invest in top-of-the-line shoes (and they will cost you steep, but it's a must in today's game), or to buy a "slide sock." A couple of different companies make them.
Jess