Being essentially lazy, the dishwasher idea appeals to me as definitely worth a try. To stay out of trouble on the 'domestic front', a couple of suggestions:
1. Do it when the wife won't be home for an hour or two.
2. Don't tell her you did it.
I'd put it in by itself, not with the dinner dishes.
Set it in vertically with a chamber dropped onto one of the racks' vertical rods.
I'll try the dishwasher soap first, on 'small load'. If that doesn't work, do it again on the 'pots and pans' cycle. I would NOT put any solvent in there!

BTW-- When I was in the Army we used to do an annual cleaning by dunking the metal bits into a barrel of boiling water with a cake of that
caustic yellow soap we used in the mess hall. That method really worked!
I've been using a 12" length of 1"x4" with a strip of terrycloth wrapped round one end and stapled on (to the back side, doh) and a strip of the finest emery-cloth on the other end. I then take the cylinder a rub it, first onto the emery w/ some solvent, then on the terry for finishing with with a drop of oil. That works on the flats, but does not get into the fiddly bits.
You can use the power tooth brush for the rest. I tried that method for a few weeks, until my teeth began to turn black....
BTW-- We're all talking about cleaning our stainless cylinders, eh?
Don't we care about our blued cylinder faces? Or is it that we just don't notice the carbon build-up on the darker-complected cylinders? If all that carbon on the faces is harmful, then equal attention should be payed to the blued guns,
but if all that gunge is purely cosmetic, then (
ach!) why all the fuss?? I'd prefer to let my revolvers look 'well-used'.
I'm thinking of just letting mine go now, for about a year, then do the boiling water method. Yeah, all that scrubbing by hand every time I shoot my revolvers is just too time-consuming, eh?