Sounds troublesome. I am lucky to lift at a crossfit gym run by guys that are cool with me just doing the oly lifts and strength training for the most part, although I am not against crossfit in the least.
I guess the only thing to do is to talk to them and explain that you really just want to do the lifts like a weightlifter, not a crossfitter.
I hurt my back awhile ago deadlifting, but your problem doesn't sound like mine in symptoms, it does sound like mine in that it probably came from rounding the back(not really sure what else it could come from) I'd read this:
and try it. I doubt a back injury is a muscle problem but I have heard the starr rehab thing often works in a black box fashion for the back. I've also read rippetoe say back extentions are good for fixing the back.
In addition to ibuprofin and contrast baths and the usual. If you're not getting good first hand coaching and you don't have previous time with heavy deadlifts you might be rounding the back too much in the quick lifts and messing it up that way.