I know what you mean, and don't know of any lube that will work in the bucket. I think waiting, or hoping, for a drier time to do the digging would be one of the only ways to do it with any success.
It reminds me of having had a similar problem, and also remember once having to dig a 6' deep ditch for a water line in wet clay. Had to attach a water line to a hydrant (the kind that drained back everytime it was shut off so the stand pipe wouldn't freeze) and it was buried 6' down.
I couldn't throw the clay off the shovel, so I used two shovels, and loaded them both with clay mud, threw the shovels out of the trench, climbed out, and cleaned off the clay mud with a scraper. Threw the shovels back into the trench, and repeated - time after time after time. Eventually finished the trench. It ran alongside a pig pen, which didn't make it any more pleasant being in the trench, and all the dirt had to be along one side so the pile was pretty high.
Wish you luck.