I think the stool is a great idea.
As for requesting its use, I would do what I do before I travel. Email sit down restaurant staffs directly about the issue on their contact us page. Since I have multiple allergies, I need to know if I can have food at a particular place, and finding out in the moment giving the restaurant no time to prepare will disappoint me and stress them. But finding out a month prior that I may have the need and seeing if they can accommodate works well (a few places can't, most have been able to do so, but I always contact prior).
It does mean less spontaneity before eating out decisions, but then it's like a Disney trip. You've made decisions on table service eating before you leave and you'd do so again now.
PS - I think with warning, 95% of places would say okay, b/c they'd literally just remove the chair before lunch or dinner start. Although I say this if you are one of the 1st mealtime diners (which I always am - it's easiest for an already clean kitchen to accommodate allergies b/c they don't have to stop, reclean, and go, and they have the most time at the start before the restaurant is full- and safer for me, too) - having the restaurant set for you with a certain spot saved (aka, the easiest one), and then adding back the chair vs having to do it on the fly in a dinner rush is probably what they'd prefer (and you would, too).