In most things in life (archaeological research seemingly not an exception) to acquire a thing (in your case - access and cooperation) - one need to define a "win-win" in the eye of the collection curator or the responsible archaeologist. Curators and archaeologist IMO want different things. I think it best for you to pause and develop essentially what could be called a classic Business Plan (as brief as it might be) to define to both yourself and OTHERS:
1) What your goals are
2) Your plan of how you will accomplish them
3) The resources you need to accomplish those goals
4) The places those resources exist and who the "gatekeepers" are
5) Their gain (win-win) from that cooperation
6) Your strategy for gaining that cooperation (advocates, Letters of Introduction, opinion-shapers, collection donors etc)
Some gatekeepers want homage. Some want tangible intellectual capital (which your results will create). Others are swayed by donor money/donor wishes. I would give thought to a multi-dimension approach to securing cooperation. If one plan fails - have a back-up. Ask senior people in the field "Who knows curator/archaeologist Bob Smith?". Try to avoid going in cold - its too easy to say know. Try to secure the advocacy of a connected person - and have a call or letter proceed your request. Be prepared to define the gain to the field/institution that comes from your work. Of course you would make a copy of your vetted work available to them and acknowledge their assistance in the work (note that in the request). Your thesis advisor(s) can probably tell you who is connected to whom. If a collection is funded by a donor - see if you can connect to that donor (without the curator feeling you went around them).
I would give some serious thought how your work can and will support related areas of archaeological work and study. Don't appear to be stand-alone and insulated. The more impactful they believe your work to be - the more likely they will be supportive of it.
Hope this helps.
Alan