Never had a problem there before, but definitely better to seek forgiveness than try to get a NPS official at Rainier to give you permission to do anything! Be respectful, leave no trace, and hopefully you'll be fine.
That's really a different issue, though if your point is that the NPS is likely to enforce the rules without paying the proper respect to us skiing royalty, then perhaps it's perfectly on point.
I thought about doing this - camping at White River - last weekend but bailed because it was raining when I got off Interglacier and the forecast turned to suck. If I were to do it, I'd pay using the self-serve kiosk and keep a lowish profile. I've absolutely seen people tented there prior to opening multiple times, for what it's worth. There are some dry tent sites in the first loop.