What finally worked (and it was still a pain) was the following:
Open a beer, set aside for now.
Insert bushings into brackets
Throw bushing/bracket assembly into ice cream bucket of boiling soapy water, let them get HOT to 'soften' them up (lol, soft being a relative term.)
Drill 1.25" hole through a section of 2x4. This will serve as clearance for the end of the sway bar to exit the other side of the bushing
Heavily grease the ridiculously oversize sway bar end that has to fit through the birth canal (ie: sway bar bushing)
Take bushing/bracket assembly out of steaming hot soapy water. Do this with a hook/screwdriver to not burn your hand (yes, THAT hot)
Set it with the exit side down on the board centered over the hole you drilled. MAKE SURE you are putting it on the right way. Triple check this because once it's on you will kick yourself because you put it on backwards. After triple checking position of bushing/bracket, double check it again.
Put the bushing/bracket back in the boiling hot water because it has now cooled off from quintuple checking the position.
Set the bushing/bracket exit side down (again) on the board over the hole you drilled.
With all your strength and weight press the sway bar down against the birth canal opening, working the end back and forth into the bushing.
Once it pops through (and the bushing did not come out of the bracket, drink half your beer.
Repeat for the other side.