domain privacy mistake, possible to fix?

So I found a good expired co uk domain and decided to register it on internetbs. At first I thought about using fake details, but during registration there was a checkbox for whois privacy which was free, so I thought why not and used my real info. But when I run a whois check on the domain, my name still shows up and only the address is hidden. Awesome.. now I can't use the domain for my pbn. Would it be possible to salvage this by moving the domain to another registrar with different account information? Or is it completely ruined already because my real name (same as on money site) has been put into the whois history?

One bad and one good thing:

The bad is that WhoIS records will always be available since the history of the domains first registration. They will thus see your information when digging. Pretty easily, to add.

Good thing is that if you would ever get in trouble for what you are doing, you can claim that you bought it and sold it to someone else (unlikely to get away with). If you want to stay entierly sure, sadly, you have to buy another domain.