I believe the key issues are: time in country, and where it was registered.
You can drive non-BG car in Bulgaria for some months, before you need to register. If you are spending time in BG & TR, then you could probably drive back and forth. However, you need a vignette (even for non-BG car), so you will be in the system... so I think you would need to be careful to make sure your vignettes matched your time in the country... otherwise it could like you have been here for years!
I think the only issue for import tax is where it was registered. EU car is OK. Non-EU car is 10% duty and 20% VAT. (I think for 10,000 euros car, it is therefore 2,000 euros of VAT, and then 1,200 euros of duty, as duty is on VAT inclusive value.)
I was at Kat this week, to register my UK car. They wanted registration document and invoice. So I think you would need to find your invoice or bill of sale from 3 years ago, and make sure it looks official (stamped if possible).
It seems a bit unfair if they base it on the invoice value of 3 years ago... so I don't know if they use this number, or discount it by x% because you've used for 3 years. I'd imagine they can look up the "book value" of your vehicle today, and base it on this (as this allows for the depreciation as you used it for 3 years, and avoids underpayment because of falsified low invoices).
UK is no longer EU, so VAT / duty currently applies.
Fortunately, I was included in the prior treatment as I drove the car here at end of 2019, and they were able to look up my first vignette is in the system to confirm it.
I had to pay 350 leva for an eco tax... plus 150 leva for new registration documents and new Bulgarian plates... plus insurance... plus a few hundred to attorney for helping me get through the system (if you speak Bulgarian you probably don't need any help). So this is separate from VAT/duty.
With Covid crisis I couldn't do the transfer last year. However, I did drive the car regularly on the UK plates, until my UK insurance expired. However, I now realize that it should have been transferred to BG plates within 6 months, but nobody stopped/questioned me. My attorney said that as I didn't register within the correct time, there is (officially) a 200 leva "late" fine.
I got my local insurance agent to issue a BG policy last week (for my VIN only, not the UK plate, with update required within 7 days once BG plate issued), so I have been driving around for several days with this new insurance (but still UK plate), and again no issues. Now I have my shiny new BG plate, so everything should be fine now.