Gluing it up became quite a challenge as there were so many fingers to try to glue.
I am wondering if you can glue up two sides at one time, let the glue dry and then continue on gluing up the sides, in stages.
You might try a glue with a longer open time.
After screwing up one glue-up, I tried Titebond Extend with a 15 minute open time versus 5 minutes for regular Titebond.
I messed up the next one, and started playing with epoxy. There are lots of options here depending on how tight your joints are and what sort of times you want. I've been playing with T-88 with a 40+ minute pot life, a couple hours open time until it gels,and gap-filling properties (it will make a strong joint with at least a .010" gap as from bad miter angles) although the high viscosity makes it hard to get invisible glue lines. It's water proof, somewhat flexible, and the stuff used to build wooden airplanes and boats. I might try West Systems or a thinner offering from System Three next. No clamping pressure is required to get a strong joint with epoxy.
Epoxy is not water soluble so you can't just wipe off the excess; you need to wear gloves to avoid developing allergies to it; it's a two-part product which needs to be mixed accurately (weighing seems easier; I got a pocket sized digital scale which reads in .01 gram increments) and stirred enough (otherwise it stays sticky forever); and it soaks in some so staining might get iffyier (but varnishes and polyurethanes are OK after you've scraped and sanded the excess).
Uncured epoxy is soluble in alcohol; maybe that could be used to wet a rag like water on yellow glue. I should try that.
White vinegar also works for cleanup. It's not toxic, emulsifies the mess, and won't carry the goo through your skin like acetone.