None of this makes any sense to me (seriously). A private instance of a blockchain? How’s that any different than a private server? Who’ll have access to this? Who’ll do the mining (or equivalent)? And why?
Edit: Let me just add that I know very little to nothing about blockchains.
Regular databases can be distributed across many servers.
If the blockchain is private, the DMV would have unilateral control to just roll things back if they desired. All of the public mirrors would notice that but they'd have no say in it. If the public nodes did have access the DMV wouldn't be able to control the chain (and wouldn't be able to revert mass errors/failures) and they won't let things out of their control.
How is that immutable if all the nodes connected to the blockchain are governed by the same entity? If you mean transparent, then ok, I can see it. However, my question then is, do we have an actual problem with transparency in car registry?