Color me astonished.
I found the Promenade parts diagram (in front of my face this entire time) and was pleasantly shocked that, rather than a canister of some sort, there's only a very normal looking flapper valve. It looks like the flush lever & rod are reversible, made even easier by the squareness of the tank.
So I'll buy this thing, take a stab at drilling a hole in it, and post back in a few weeks with pics of my work. With a voided Toto warranty, of course!