I am unclear on how you are capturing your fish waste solids during normal operation. If you have an external pump, be pulling the water from the center bottom of the tank would be ideal. This requires keeping your tank water in circulation by returning some of the water from the pump back into the tank in such a way that it circulates the water. Your pump size should be adequate as long as that rating is at the height of the grow bed above the fish tank water level. Some pump's flows are rated at zero pressure and some are rated at one meter of head and some don't say.

The other thing I'm not clear on is the shape of your fish tank. I am assuming it is either oval or round. Oval is adequate to allow for circulating water. We use a 29 gallon rectangular aquarium for our indoor aquaponics system and it does not accumulate solids. It has ample jet back from the 360 GPH submersible pump located at one end, and along with the flood and drain from the 25 gallon grow bed, it keeps the water stirred up enough the keep the solids suspended for the pump to grab them.

The numbers you give for your water pump look good for the size of system you are running, but it is the configuration that I do not have a good picture of in order to fully evaluate what is going on. The number of air stones doesn't mean nearly as much as the amount of air your air pump is delivering through them.

Four times an hour is adequate cycling times.

Pictures would be of help.

Oliver