“The Temple Mount is a unique place for prayer,” the letter said. “As Prime Minister, we ask you to do what is necessary to establish a place of prayer for Jews on the Temple Mount, in a respectable and peaceful manner, that will honor the holy place we all long for.”
I've got the spot. I can be approached without going necessarily through any of the existing gates and can be actually relatively out-of-view.
To explain, here is a view from the east:
The arrow is pointing to the colonnaded walkway just north (to the left) of the Mugrahbi Gate entrance. It is actually, in part, after the exposed Western Wall disappears behind the Makhkama Building, the eastern section of that building which houses the special Border Guard units.
In other words, entry can be facilitated by opening up a doorway there.
The colonnaded area
could be closed off using heavy clear plastic that can be affixed with a one-way view, outward. That way, outraged Muslims cannot even see the Jews inside.
Another possible entry point would be through the underground Barclay Gate and up the stairs,
In is not that large an area but could accommodate probably 100 worshippers, depending how far south or north the area closed off.
The walkway is rarely used, if at all, since it leads nowhere but ends where a library is located.
A view from above:
From a Halachic point of view, the Western side is relatively the most holy as the inner courtyard of the original Temple precinct was closest to the west but since the area to be set aside is quite narrow, about 4 meters at the most, that would pose no problems.
The one problem is that if the synagogue does receive government permission (don't worry, it won't), that would enable the government to limit religious Jews only to that area and the encircling of the Mount as done today would be terminated.