The question refers to the Speaker of the House as Commons as "a single keyholder of three to the Palace of Westminster" - but this is not quite right.
The three keyholders referred to here are the Speaker of the House of Commons, the Lord Speaker (representing the House of Lords), and the Lord Chamberlain (representing the Queen). However, they only have joint control of Westminster Hall and the Crypt Chapel (sources: 1, 2).
Of these two particular parts of the Palace of Westminster, only Westminster Hall is used for addresses to both Houses. So, the Speaker can indeed prevent anyone from addressing both Houses in this space.
However, the Royal Gallery has also been used on a number of occasions for this purpose - and this is under the control of the Lord Chamberlain alone. But although the Speaker cannot prevent an address to both Houses happening here, it seems unlikely that that would happen without his agreement. As he said in his statement:
Customarily an invitation to a visiting leader to deliver an address there would be issued in the names of the two Speakers. I would not wish to issue an invitation to President Trump to speak in the Royal Gallery.
Finally, as to the question of whether the Speaker will "be able to prevail against the Prime Minister": yes, because the Government has no power over the use of the Palace of Westminster.