Off the wall thought here. There are some small handheld waterproof misting fans that are pretty cheap. While the misting feature would not be of any help, I wonder if the mechanisms are sealed well enough that you could use the misting area to counterweight it down, and just drop it into the tank with it turned on. The fan blade mechanism would agitate the liquid inside the backpack and be safe as long as the battery compartment and the blade bearings seal out water penetration. Even if the battery compartment is not sealed, you could put a little silicone around that area and peel it away when the batteries need to be changed.
These things are real cheap, many under $10. Just get one that has a shroud around the blades so it operates continuously. It would probably be easy to even try a little velcro with double sided tape to attach it in a particular spot at the bottom of the tank and thus be removable.
Crazy but might just work.