I'm presuming that, because of Thrain's rank, Sauron dealt with him himself; so he took the Ring off him, and didn't bother with anything else, this being the specific item he wanted. He then left Thrain to die.
With that in mind, I think that Sauron's lesser servants would have then left Thrain alone, that being what the Big Boss had done.
After all, there are far less repressive organisations in which people take a lead from the top person's behaviour.
I'm also presuming that Thrain held on in the
very remote possibility that someone from outside might come in, to whom he could give the map and key. That happened in the case of Gandalf; so when Thrain gave him the map and key, he was able to die having, in this small way, done his duty as the head of the House of Durin. As
The Hobbit later showed, this devotion to duty paid off.
