Order is really up to you and if the plumber doesn’t mind doing multiple hits, if he can then do what I said, tile so much and tile down to the bath. I’ve always filled the bath, mark the wall level and fixed a batten to the wall, loads of silicone then drop the bath onto it, it won’t really bare any weight but it will as much as possible stop the bath moving up and down when full/empty.
Have you used these 6mm Hardiebacker boards, I have on the advice of a useless plumber, I will be pulling these up quite soon and replacing with 15mm ply that I wanted to use in the first place. The backer boards are flexing like crazy.
I presume you mean putting the tile backer board onto a structural board, not sure 6mm alone will span between joists – but happily proven wrong. No more ply – is also meant to be an alternative to……well ply.
When glued and screwed to solid floorboards it’s perfectly rigid in my experience.