I was generally aware of the situation, thanks to earlier articles from you, but I was still shocked by the numbers in the last paragraph.
There is no real remedy against coins that have already been stolen. Some may be recovered by chance every time the political winds change, but most will remain lost. The point is now to stop the thefts. This can only be done by the Bulgarian authorities and they have to be motivated to do so.
What should have been a natural starting point to build such motivation, UNESCO, is a pretty useless organisation. My advice would be to use the EU instead. That will be difficult, as the EU has no cultural mandate, but it can be done. Bulgaria is a recipient of regional and structural EU aid. I would expect it to be largely for infrastructural projects (roads, bridges etc.), but there are bound to be tourism projects. Make sure of that on the vast EU web site. These can be tied in to museums and from there you can go to theft prevention, safety and stock taking/registry, perhaps even with EU subsidies. Since there is money and politics involved, the old guard cannot resist a deal where they need to open up collections for the politicians to get subsidies from Brussels.
That shifts the motivation question to Brussels, where you can achieve stuff the usual way. By lobbying. You and Mr. Paunov, being outside Bulgaria, are well placed to lead. Get at least the top of the university interested in your story. Because of Brexit and increasing British irrelevancy in Brussels, you will want to get the Irish involved next. Some kind of Celtic event (there is one in Normandy annually, centred on music) may give you an opening to the Irish powers that be. An interested parliamentarian or culture 'crat could open the door to the taoiseach. Otherwise, one of my old contacts may get you an appointment at least within his office (but get Cardiff university squarely behind you first). They, in turn, may get the French along, playing the Celtic card; the French are quite sensitive to minority interests. Together, they can move the Brussels 'crats to actually do something.
I am sure there are other paths. My point is simply that you have done what you could do in the scientific world. Now, a step into the world of politics is needed. The way sketched above is not the only one. Just keep an open mind, use every opportunity you may get, especially when you can play on minority interests. You should keep your eye on the goal, make progress, not noise and play it calm and with dignity. Have faith in your story and never lose hope. The final comfort is that the old guard dies before you do.