I expect that decisions on who gets exhumed, and who doesn't, reflects the
same sort of political and social necessities that underlie the timing of
canonizations (in the Middle Ages as well as nowadays). That is to say,
unless many 'constituents' would gain (spiritually, and maybe even -- in the
best sense -- financially and politically) from the publicity surrounding an
exhumation and study of whatever remains may be found, then I think that no
amount of scholarly argument could possibly convince 'the authorities' to
allow an exhumation to proceed.
And unless I had many, MANY cards in my hand -- e.g. qualifications and
experience in the realms of medieval archaeology AND historical pathology --
personally, I wouldn't waste anyone's time trying to get permission to
exhume a pope (or anyone, for that matter).
Who to ask for permission, if one were a renowned archaeologist and
pathologist (and with good connections)? I would expect that possession
accounts for almost all the legal power in such matters, so I would write to
the person responsible for the cemetery or church where said person/pope is
buried. But again, I really wouldn't bother anyone without demonstrably
being suitably qualified.