Tiling over lathe outside wall with window

mkjeld

I am new here and posted this first in construction before seeing these other categories.

Hello all - I am new to this forum. Have renovated a total of 10 houses, but just basic stuff. Now I am working on my own house and really need help - had problems with contractor showing up etc so decided to finish the project myself.

Adding a 2nd story bathroom in 100 year old farmhouse. The bathtub will sit in a alcove on an outside wall. The ceiling above the tub is slanted, so the wall height at that side of the house is about 5'6". There is a window on this wall that sits at floor level since it is a short wall The walls are lathe & plaster and the plaster has already been removed. I will tile around the tub.

I don't want to remove the window because I do not want to change the balance on the outside of the house. We left the storms on outside, took out the inside windows and hung a blind there, then we framed it inside, put in white foamboard for insulation then osb board over that, so the window is covered from the inside. The window frame sticks out from the lathe so we need furring strips to level the wall with the frame. After furring, do I install the cement board directly to the furring at the studs, or should I install some osb or plywood first? The wall has blown in insulation. We are in Iowa and have cold winters here. I also wanted to put a skylight over the tub since the room will have no outside window for light.