The Temple Institute is an organization that is preparing for the arrival of Mashiach. They are preparing the different vessels to use in the Beis HaMikdash, as well as the materials to use in building it. They have received brachos from some major Rabbonim, including R’ Ovadia Yosef zt”l, R’ Mordechai Eliyahu zt”l, the previous Bostoner Rebbe zt”l, and others. Every year before Pesach, they prepare a lamb in case Mashiach comes just before Pesach, and a korbon Pesach is needed. They usually bring it to R’ Chaim Kanievsky to be inspected, and he gives them a brachah and thanks them for the work they do.
Yes, they do hold it is muttar to go on certain areas of the Har HaBayis, with the proper preparations. This is in line with certain Rabbonim who hold that way. It should be noted that R’ Moshe Feinstein, in OC 2:113, seems to hold it was ok to go on the Har HaBayis. The Teshuva addressed a question raised regarding the fact that both the Tur and Shulchan Aruch it makes no mention of an issur to spit on the Har HaBayis in our times (or in their times, I guess.) The question was asked, is this because we are all tamei meis and can’t go there at all? R’ Moshe refutes this, and gives an alternate answer, which does not involve an issur to go on the Har HaBayis at all. R’ Tendler shlita has said numerous times that his father in law, R’ Moshe, never tried to stop him from visiting the Har HaBayis.
If your Rav holds it’s assur, don’t go! But don’t insult those who follow their own Rabbonim who are mattir it.