I've got a registered AR I take to the local county-run range. I just haul it out and start blasting. They've never asked me for papers. (I've got a xerox in the pistol grip just in case.)
Other people might have different experiences at different ranges, and some people may be risk-adverse. YMMV.
Presumably they would need probable cause about something you're doing in order to accost you. The simple fact that you are carrying an AR would not be enough. However, probable cause is pretty easy to manufacture if the cop is so inclined, so it may that they see you at the range, want to run the rifle, and see that you've got an unbuckled seatbelt when you drive away (from 200 yds on a dark, moonless night in driving rain). Once they have you pulled over they ask about weapons, run the serial numbers, and discover that that one isn't in the database.
It wouldn't be worth the risk to me. If separation anxiety was a big deal I'd drive to AZ or NV and shoot once in a while. It's a maximum of about 4 hours from anywhere in CA. Make a day, or a weekend, of it.