What Bulgaria accepts from Scotland
Bulgaria takes a single-step certificate added in the United Kingdom. That mark is the whole of what a reader in Sofia looks for, and there is no slot on the page for a second one. Consular legalization for Bulgaria is therefore not part of this route, and paying for it would add cost without adding any weight to the file.
The change is easy to miss, because Bulgarian offices ask for well stamped paper and say so firmly. What they want is the register entry itself, signed and certified. A Scottish extract that carries the British certificate is complete as it stands, and a row of extra marks would not make a clerk in Plovdiv any happier with it.
Where the Edinburgh paperwork starts
Only one British desk matters here. The FCDO Legalisation Office adds the certificate and runs the job by post from PO Box 6255, Milton Keynes MK10 1XX, so there is no counter to visit and nothing shifts because your address is in Edinburgh. What does change is where the paper starts its journey.
Scotland keeps its own civil register, so a Scottish birth or marriage entry is not ordered the way one in England is. Edinburgh Registration Service, City Chambers, 253 High Street, Edinburgh EH1 1YP, issues certified extracts of those entries. An extract travels well because a lost one can be reordered in days, while the certificate you already hold cannot be replaced so easily.
Cyrillic, and the name on the page
The real risk on a Bulgarian file is not the stamp; it is the name. Bulgarian runs in Cyrillic, so every Latin name is transliterated by hand at the receiving desk. Two clerks can write the same Scottish surname two ways, and a bank or a registry will then treat those as two different people. Settle the spelling before the file leaves you.
Keep one spelling across the extract, the Bulgarian text and your passport, and state plainly which one is right. Where a marriage entry carries both a maiden name and a married name, both belong in the Bulgarian version. Embassy legalization for Bulgaria from Edinburgh would not have solved any of this, because a consular stamp vouches for signatures, never for spellings.