Not a good plan. In a typical, conventional, shower, there are weep holes under the surface to allow any moisture that does make it below to weep out and down the drain. Installing a new drain like that just wouldn't work well, even if you could find something to fit. Also, bituthene is somewhat flexible, and not designed for thinset to adhere to, so the combination of the two is problematic.
Depending on the method of construction (hotmopped is common in CA area), while it may not be leaking now, that doesn't last forever. A tearout of the pan is a better plan.
IF you can raise the existing drain (it's often threaded but after 20-years and the mortar and thinset it may not move), and you cleaned things off well, if you have confidence in the longevity of the pan, you could tile over it. Some nice porcelains are barely 1/8-inch thick (I used one like this for the floor of my mother's shower), and you may be able to tweak the drain cover to accommodate it. It's rarely a good idea to patch over an existing shower if you want it to last another 20-years.