1" subfloor thickness is considered minimum for tile installation. This can be achieved with 1/4" Hardibacker over the plywood. However, in you case the problem around the floor register is something else, and adding extra floor may not be enough. I would consider opening up the ceiling to see if you can box the area in with some 2X8 material.
Between the backer board and the tile, you just have to suck up the fact that you will need to deal with issues like the flange, the doors, etc. Happens all the time.