Best would be to open the tank, take out to reel, and put either only the film, or the reel with film, on the stabilizer.

Some people warn that stabilizer is no good for your reel, but you can clean your reels with toothbrush etc.
The funnel and the light labyrinth of the tank are not so easy to clean inside. And you want to avoid any contamination by final rinse or stabilizer. So I would avoid having the tank getting in touch with the stabilizer. Your mileage may vary.