AFAIK you have a receiver, ESC and ballast tank control board. Anyhting else?
The ESC is a fairly bog standard design, I believe with a built in BEC.
From what I've sen of the control board, it looks like a knock-off of Norbert Bruggens design. A micro controller based relay switcher, that simply enables the peristaltic pump to push each way. I believe it has a low voltage failsafe, missing pulse detection, a pressure senser input to cut the pump before the bag bursts (hopefully), and maybe a water detector(?).
If the receiver is a bit duff, it's entirely possible that you would experience the symptoms you are getting. Add to this the fact that you have tried it with a separate TX, and it is still playing up.
The other thing I would check is to ensure you are getting a good 5 volts to the receiver bus from the BEC. Ideally this wants to be done with a load put on it. You could wire a 5 ohm 10 watt resistor across the leads. This will pull 1 amp across the BEC, which it should be able to manage. Measure the voltage, it should maintain the unloaded reading, if it drops significantly, you might want to consider an external BEC. Depending on the make of receiver, some will happily work down well down to about 3 volts or so (i.e. a lot of futaba kit) whilst others will start to flake out if you drop much below 5 volts.
One other final thing to try. Change the crystals in both TX and RX- sometimes they can get damaged, and I have read that they age, although I'm using crystals that are years old without issues.