This was a bit of a challenge for me, how to move this onto its concrete socket without destroying the tank, the socket, and my back.
After much pondering I figured out a solution, and with a couple of uninterrupted hours and some good luck it worked pretty well. I temporarily attached a 2x4 board to each side of the tank, trying to hit the center pretty well. This board would extend beyond the concrete socket and give me something to work with and lift it up.
For some reason I have a lot of chopped up karate boards in my garage
and they came in very handy. After pushing the tank in place, I raised it up, one by one, supporting all four corners, and using the center boards to push it up and up and up. Once at the right height, I pushed it all the way to the wall.
Some magic, and bam, it's floating in air!! Not quite that easy, but the 2x4's did what I thought they would do, and gave me some space to add a layer of thinset. Lowered the tank, pushed it all the way to the wall, making sure it stayed fairly level. Worked out pretty well. I filled in the gap where concrete was missing, and now it already looks like it's always been there.