in my opinion backer board is the way to go it will not shrink rot or take in moisture and it will not create a step at the door. a quote from BS5385 3.2.4.2
The use of sheets and boards that are subject to movement from changes in moisture content, e.g.
wood-based materials such as plywood, chipboard, wood particle boards, etc. should be avoided if at all
possible. If such boards have to be used they should be restricted to small areas and tiles should not bridge
joints between boards. It is good practice to seal all exposed edges and the backs, but not the faces, of such
boards with a suitable sealer to prevent distortion by atmospheric humidity changes. Care should be taken
to ensure such boards are not installed in a condition where their moisture content is higher than the
ambient equilibrium moisture content once the tiled installation is in use. Failure to observe this can lead
to subsequent warping and distortion of the boards with consequent cracking and delamination of the
tiling.