Still have the issue of no one wanting to mine it, leaving all the mining work to the government. Then there's the issue of Bitcoin type networks being global, meaning the government's mining activities could be supplanted or screwed with by foreigners. About the only secure option a small government would have is to either force everyone else in the country to mine, too, or get rid of mining and just run a secure ledger all by themselves, having total control of all coin distribution, and tracking all transactions digitally from a single centralized location (what central banks already do).
I don't think there is a way for any government or private entity to own a private Bitcoin clone, simply because not enough people would ever be willing to mine/secure it.