Hi DIY SG,
I would use 3/4"t&g plywood. Screw it down. I like to use decking screws.
The 1/4" hardi backer board will work if you install it in thin set. Use good quality thin set...Like Mapei or Latacrete..and not from a big box store. Again, you need to put some screws in the cement board to make sure it is well bonded to the plywood. I take it that the heat will be the electric tile warmers? It can be installed on top of the cement board.
It depends on how bad the damage is on the joist...If it is...then you will be fine with what you are planning.
Also...I would insulate the floor while you have the joist exposed...If it is on the second floor.