The way I've seen them done is the pan is set directly on the floor & the mortar is placed on top of the pan, sloped toward the drain and the tile set on top of the mortar.
Even if water gets through the grout, it shouldn't be leaking through the floor.
Caulking the tile is a fix, but not a good fix. Eventually it will fail, plus you have wet mortar under the tile that is going to grow things.
I would plan on eventually removing the tile and fixing the drain.
Before you caulk I would pour some bleach into the cracks and wait a day or two.