We're going to have to replace them because NYC is mandating a backflow inhibitor which will cut the water pressure a lot, and they won't work. (Or, if they do work, they'll have much too negative an effect on the apartments above us.)

Other alternatives are to install costly pressure pumps and holding tanks, etc. Which is not really a desirable step.

Replacing the toilet is going to be problematic for several reasons:
-it's on an 8" rough-in (so, ripping up the tiled floor)
-it's got a side-intake flushometer (so, ripping open the tiled wall perhaps)
-it sticks out from the wall about 20", leaving plenty of room to stand in a room that's 50" wide; I can't find a toilet w/ less than 25-1/4" inches

So I got to wondering--can we just mount a tank above the toilet on the wall? You can buy the high tank by itself; can we simply attach it to the flushometer toilet?

And if so, can we offset the tank by a foot or so, or does it have to be directly below?

You can not install a high tank.

Based upon the age of your bowl it requires either 5gpf or possibly 3.5gpf, but code now mandates that you cannot install anything bigger than 1.6gpf, which would not be enough water to flush your bowl.

Any attempt to modify your bowl would be classified as "New Work" which means you would be required to install a 1.6gpf watercloset.

So it's the design of the bowl itself that requires us to have that much water?