As things stand at the present time, getting a Bulgarian Passport would necessitate giving up the UK passport. In my case it would also involve a wait of five years. While I could quite easily pass as Scottish, I could never really pass as Bulgarian, no matter how well I perfected my language skills and accent. Perhaps I could change my name, but I wouldn't really want to do that. I tend to think of myself as a creature that lives on Planet Earth; but you have to have a passport or you are stuck.
I would guess that a Kingdom of Scotland passport would be a successor to the UK passport, as would the Kingdom of England and Wales passport, and that they would both carry the same international clout, but the Scottish passport would have the added clout of being part of the EU, while retaining the right to return to the Isle of Britain at some future date should one wish to do this. If you have only a Bulgarian passport this may not be possible.