See my rather ancient post here:-
viewtopic.php?f=31&t=18335&p=365568&hilit=residency#p365568 The white card was superceded by the photocard that many have now.
I don't think much has changed (yet) and a UK EHIC card (if you have one) is, I believe, still being taken as proof that as a resident you will have medical insurance in case of illness. This is incorrect as the EHIC card is intended for non-pensioner holidaymakers to have medical cover in the event of a medical emergency when in BG. The alternative is to buy private medical insurance, which may not be cheap. But if Bulgaria MBR recognise the EHIC as giving you medical insurance, what should you do...?

If you don't own a house, then a copy of a Bulgarian property rental agreement will be needed. I also needed to show some proof of my company registration as in those days (up until 2012) individuals from within the EU were not permitted to own a house with land. (the land only was the problem) and it was necessary to set up a company to own a house and land. That rule does not now apply, but I chose to keep the company-owned-house arrangement in place as I suspected the future BREXIT vote might cause problems....
Do not forget that interpretation of the lichna karte rules can be very variable, depending on which MBR office you visit. Unfortunately, going along and asking them what you need beforehand is also no guarantee of success as they may add in 'extras' when you actually try to make the application. Also take along one or two photocopies of everything that you have.....
Let me know if you need any more help/advice.