So I went tonight, sat at the counter and asked about attempting this. It immediately turned into a classic Angel/Devil situation: my server was trying to tell me all the reasons not to do it, while one of the girls behind the counter was chanting "Do it! Do it!"
After some reasonable discussion, we came to an inescapable conclusion: at the pace I'd need to eat in order to not get sick, there'd be no way to prevent the sundae from melting into an ice cream soup long before I finished. So I settled for the Mickey Pants Sink, which they were more than happy to make in a Chocolate Lover's version for me, and finished that off while it was still (mostly) solid. Plus I got what will likely become a fun new food dish for my cats out of it.