Regenerative turbine pumps work well, but the necessary close clearances dictate only clean fluids. Gear or rotary positive displacement pumps work well, but they have slippage problems in low viscosity service and their very low capacities may not be sufficient for the application. Metering pumps are good for very low flow, but the inherent pulsations can damage some instrumentation. You can connect single stage centrifugal pumps in series if a single pump cannot meet the head requirements. This pump was developed during world war II to handle the high head low flow rate requirements of the German ram jet fuel pump.
Internal recirculation can overheat the volute and cause cavitation problems.
A high differential pressure across the pump can damage close internal clearances. The power loss can be expensive. The increase in stuffing box temperature can cause a premature seal failure. Here is the proper way to vent a centrifugal pump after it has been installed, or the system has been opened. I am assuming the pump is empty of liquid and both the suction and discharge valves are shut.