The rivets are not just an old car/Mopar thing. the same rosette rivets are used across all car brands, up to current, I believe (I only have 2008/2010 vehicles handy to verify).
I have yet to have a traffic stop where the vin was not checked against the registration- they need to confirm that the plates are on the car they are registered to.
And the OP is in California, not Wisconsin. They can require VIN verification at time of registration in most cases, and is always required if the car is brought in from another state. Although it doesn't apply in his case, any car 1976 or newer goes through a vehicle inspection every two years, in addition to smog testing.
Since the right rivets were found for under $10, just do it right and avoid the potential headaches.
Heck, last time I had to strip the tags off a dash, I requested a State Patrol officer to inspect the vehicle in question, and he GAVE me a set of rivets to reinstall the tags with. (Yes, they have them.)