1. Don't run the pump
2. Separate the flat shaft from the slot
3. Use a round shaft
4. Seek professional help

This sounds like a low-tech coupling.  We have a setup like this that drives the main oil pump for a turbine-gearbox-pump train.  The end of the oil pump shaft has a flat end that fits into a slot in a piece mounted to the end of the pinion shaft in the gearbox.  This "coupling" was supposed to be lubricated by the oil leaking out of the journal bearing on the pinion shaft. However, this didn't provide enough oil and the "coupling" would wear out quickly. We drilled a new connection in to the lube oil system and ran a piece of 1/4" tubing up to the area of this "coupling". We drilled a hole in the housing and installed a small nozzle to jet a stream of oil directly onto the "coupling".  We did this about 5 years ago and haven't had a failure since.  If this was an option, you would need to make sure the oil system had adequate excess capacity to handle the additional flow. For our small "coupling" we sized the jet to provide 0.25 gpm or less so there was no problem with the oil system capacity.

Alignment....... If it is perfect your flat shaft will have no sliding action and thus no wear
Good Luck
Ralph

If you are using a positive displacement pump, pressure pulsation could be contributing to your problem.  Flow from a positive displacement pump is not continuous.  It fluctuates about a mean flow.  As a result, the discharge pressure pulses at a frequency proportional to the drive speed.  Normally the pressure pulses are not large because the flow fluctuation is not large, but, if the pulsation excites a resonance in your system, the pressure pulses can be very large and damaging.  If this is part of your problem, you may be able to eliminate it with an in-line suppressor.

Another contributor could be shaft hardness.  If the shaft is soft it will wear out quickly.