I'm not a professional tile person, but I'll stick my neck out and go against the grain on this one anyway.
I would be very tempted to skip the backer board, and simply apply new tile to a layer of modified thinset on properly-prepared existing tiles. One of my work buddies years ago (too many moves since then, can't remember which buddy) did it on his kitchen backsplash, and I don't think he ever had a problem with it. It's important to make absolutely sure that the existing tile are solidly attached to the walls, everywhere. A good way to check for delaminations is to lightly tap each tile with a small tack hammer, moving around in a random pattern, while lightly holding (and moving) the fingertips of your other hand on the same tile. Unbonded tiles should be easy to find, giving off a totally different sound (much "duller"), and will feel different to the fingertips as well.
If your tiles are all sound, with no "holidays" (large areas missing adhesive or thinset on the backs), your next step would be to lightly roughen the faces. You could use a belt sander, but I'd be inclined to use a small angle grinder with a masonry wheel or disk. Keep it moving, as there's a tendency for the tile to glow red and break from the heat if kept in one place too long. This will be a messy operation, so wear a mask, and keep the door closed. If the room has a window, a box fan pushing the dusty stuff out through the open window will help minimize dust in the rest of the house. If no window, turn on the exhaust fan. And tape all of the door frame gaps with wide painter's tape.
Work the thinset into the existing tiles' faces by pushing down hard with the trowel's smooth edge first; then reverse it and apply the "toothed" pattern to the mud. I always butter the back of each new tile I set, and like to uniformly tap each tile with a rubber mallet to help make sure it's bonded. Tile spacers at the corners will help keep things in place as you work your way around.
If any tile professionals disagree with any of the foregoing, please chime in and set me straight.