The Rabbi's final remarks require further examination, because several halakhic decisors wrote that it is permitted to pray in a mosque even though they invented commandments and a prophet out of their own hearts.

I know that view, and indeed that is the opinion of most halakhic decisors. But in my humble opinion one should not distinguish, as I explained. Takhen brought this in Tzitz Eliezer in the name of the Ran and ruled accordingly.
Right now I don't have a computer, so I'll quote from a lesson by Rabbi Ohad Fixler:
In his remarks, Rabbi Ovadia cites the words of the Tzitz Eliezer, who disagrees with him. Rabbi Waldenberg brings the Ran's approach in Sanhedrin (61b), who wrote that although the Muslims do not turn Muhammad into a deity, since they bow before him they have the status of idol worship. Following these remarks, the Tzitz Eliezer writes:

"Thus we have learned explicitly from the words of the Ran that even the worship of the Ishmaelites and their bowing to their false prophet has the status of idol worship regarding their sanctuaries in every respect, and if so it is clear according to this that their mosque as well, where they perform their worship, also has the status of a house of idol worship in every respect, and it is forbidden to enter there…" (ibid.).

It seems to me that he does not mean to say that Islam is not idol worship, but rather my line of reasoning.