Properly glued acrylic should not "fall apart". We're not talking about a small surface area here. Something like Weldon 4 should not fail. Using something like Weldon 4 and not getting any bubbles is a bit more tricky.
I'm working on starting doing this type of sign, but I can say with 100% certainty that it's going to cost a heck of a lot more to do it out of 1 piece instead of doing flanges or applying to a backer board. Your sheet yield is going to drop dramatically doing it this way and you're machining time will increase as well. It really boils down to preference, skill set, and to a lesser extent, economics when deciding whether or not to do a flange or a backer board from what I can tell.
Also, take this with a grain of salt, but maybe you can use gravity to your advantage. If you install the box with a slight pitch, gravity will help keep the letters in the box. Obviously I'm talking about a slight pitch as anything more than that would be noticeable and potentially look really bad.