60Hz "click" or clipping noise when differential amplifier is surrounded by salt water
Sound travels very well through water - try isolating the tank sonically or mechanically from whatever you can. This could be as simple as putting a dampening material (carpet!) under the tank. It is a long shot, but I think it warrants an answer rather than a comment nonetheless.

I once wasted a day trying to fault-find a sensitive differential amplifier circuit that was exhibiting almost exactly what your tests are showing: a ~1-2kHz single sinusoidal pulse every 20ms (or the 60Hz equivalent in your case), but no actual mains frequency sinusoid. We tried turning the lights off in the lab, the computers, equipment in neighbouring rooms, everything. We tried laptops, all manner of electrical shielding - we put a grounded tinfoil 'hat' over the circuit board - and even resorted to working by candlelight once it got dark.

When we knew we had lost the battle my tutor rested his head on the desk in defeat, with his ear to the surface, and heard the spurious signal; the desk was vibrating. We determined that the building's cooling/heating system was the source of the excitement (pun intended); it was roughly beneath the lab and the noise could be heard faintly in the corridor outside its room. A heart-warming story.

In the end someone put a scarf underneath the circuit and the interference was gone instantly. Even with the nearby electrical equipment turned back on, mechanical isolation prevented any unwanted signals.