You are not obligated in other mitzvot while you are entirely engaged in another one. However, according to the Rambam the exception is for talmud Torah.
So yes, he was obligated to do any of the mitzvot that he was commanded to do at that time. However, I'm not aware of any commandments that he missed out on.
Further, it says in Bamidbar that Gd sent a holy angel to help guide the Jewish people through the desert. Many commentators say that this refers to Moshe Rabeinu, and yet Moshe was still obligated to do his part in the Mishkan, so we see that Humans who are compared to angels are still required to do mitzvot, just as angels that are now human are not required to do mitzvot.
Lastly, the gemora says that one should not depart from the Minhag of the place they are in. However, your question is asking about changing what mitzvot you are obligated to keep or not. I think its very important (especially now adays) to not confuse Minhag with Mitzvah. Even if all of the above was not true, Moshe would still be obligated in the Mitzvot he was obligated in, regardless of where he was. Minhagim, is a different issue all together.