Depends on where on the base they'll be working. We had base housing adjacent to the secure area and if he's working on the houses they won't care. If he wants to go into the secure area he definitely needs to Uber in from somewhere else.
Uber wouldn't have permission to come in. Secure parts of bases require CAC cards (ATM machine!) for all vehicle occupants (which was fun when generals would come in, have their cac card scanned and was good... but the driver wouldn't scan good [didn't have clearance to come in]. "Sorry sir, you're going to have to walk, or wait for the Sergeant of the Guard to show up with his golf cart... Your driver needs to leave IMMEDIATELY."). When standing gate guard on a secure airfield I never cared about the vehicle plates. It was always the CAC card and if the under car scanner went off. The validity of your car's registration didn't matter in the slightest to us.
Although that may have changed at this point... I've been out for a while.
The main post guards probably would care.
Edit: Oh funny side story. That's the one time I was able to draw my own rifle on a General too... He was coming in to speak to some Colonel, but never got the clearance (probably was just visiting post and wasn't from there). He got uppity... I did what the Rule of Engagement specified I do (which does lead to eventually point my rifle at the driver and/or engine block and if they still fail to comply... shoot). Specialist me just doing my duty. He finally got the message after placing a few phone calls acting like I was going to jail... Was interesting to see that amount of ego on someone who probably orders his own people to do the same shit. I reenlisted for 3 more years after this trying another unit/job out to see if the Army was really that dumb... Out of the 4-5 units I was in... only 1 was worthwhile IMO. It basically is that dumb... Mad respect to those who deploy and do the job, I know that it sucks (as it sucked for me). But the leadership... man the leadership could use some pruning. Make it worthwhile for the people that are actually good to stick around rather than letting dumb asses fail upwards so much.
They will absolutely care about your plates. They usually check for valid registration and stuff here too, and anything else. You don't really get on a military base without being identified.
Just hop the fence, it'll be fine. Be sure to inform the Belgian Malinois flying through the air at Mach Fuck You that he doesn't have jurisdiction, you'll definitely be leaving with all of your arms still attached.