I posted your question, yesterday, on dinonline. This is their answer:
There are a number of potential problems:
Cooking: it must be verified if the cheese was actually fully cooked
in the production process. If not, bringing it to a temperature of Yad
Soledes [about 110F] will violate the melacha of cooking.
Chazara: Even if there is no cooking involved, one may not put even
cooked food directly on the fire or “blech” on Shabbos, because of the
Rabbinic prohibition of Chazara. This could be circumvented by putting
it on top of a pot or pan which is on the blech.
Melting: There is a Rabbinic prohibition of Molid, turning a solid
substance into a liquid. Melting solid cheese may involve this
prohibition, depending on how melted it will get.
Keep in mind, that their answer is one opinion, and since this is a short Q&A, the rav frequently does not source his info. There may be leniencies, esp. regarding the melting problem.
As for item #3, my thinking is that this doesn't apply to pasteurization. As far as I am aware, one cannot cook cold milk on Shabbat. Since you raised the issue, I'll see if I can research more on that topic.