I believe it has to be immigration rather than police station.
We are in the Brexit "transition period", and until 31 December 2020 UK citizens are treated as EU Citizens for immigration purposes, which makes it's much easier. On January 1, 2021 you would be treated as a non-EU citizen, and the process for getting a Residence Permit becomes significantly more difficult.
As an EU Citizen you should be issued with a Long Term Residence Permit valid for 5 years.
- You need a proof of residence in Bulgaria, in the form of a long term rental contract (in Bulgarian, for a Bulgarian property) or your Notary Act (deed) for your apartment in Bansko.
- You need proof of medical coverage, which can be a Bulgarian "Medical Insurance for Foreigners" policy which is issued immediately at most insurance offices for about 80 euros. I don't know if all immigration offices accept your UK EHIC card, but certainly mine (in Plovdiv) did. So I would certainly get/bring this if possible.
- You need proof of finance. This used to be a BG bank account with a couple of thousand leva, but not sure if this is still required. I think they will now accept a declaration and perhaps a debit/credit card. The declaration is in Bulgarian, and you have to notarize it. For my renewal (a couple of weeks ago) I submitted a declaration and a copy of my UK and BG debit cards.
- There is an application form too, of course, and I'd bring multiple copies of your passport and supporting documents.
The process (if all goes well) is that you stand in line for a few hours to submit your application... then come back and stand in line for another few hours to get your photo and fingerprints taken... then come back and stand in line again for another few hours to collect your card. :-)
At Plovdiv, this took me 3 business days (Wed/Thurs/Fri) but it's possible in 2 business days if you get there early on Day 1 and charm them into doing your photo/fingerprints same day.
If you're lucky and you get a friendly bureaucrat they may speak a little English, or they may be happy to help you via Google Translate. But I would not bet on it. So I think it's safer to find a translator / Bulgarian helper who will help with the application and declaration and navigating bureaucracy (and hopefully when/how you can avoid a few hours standing in line).
I decided to pay my attorney in Plovdiv to do it. He charges 1,000 euros if he provides the Bulgarian rental contract, or 500 euros if you have a Notary Act (as I did, and you do). As you're short of time, it might be 500 euros well spent to make sure you have all documents 100% correct, and you can fly in for just 3 days and get it done.