Eastern Bulgaria has the biggest appeal to me. I highly doubt we would fit the specialist criteria, but would Thracian tomb of Kazanlak offer an opportunity for our special access idea?
Just to throw out additional ideas, Western Turkey? Easy access to airports and a recent glut of new sites that people may not have picked up on previous visits, and don't we know someone in the area that specialises in organising tours of World Heritage Sites, perhaps that could get some specialist access aspects covered?
I'll be honest St Kilda doesn't really float my boat (pun partially intended) Entirely personal, but the amount of time and effort needed, all in my home country, and it probably not being a site the whole family could visit would put me off. Not to say it shouldn't be an option, just that if we go for it I probably wouldn't join, maybe I'll just have an evening or two in Edinburgh.