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You need to pressure both hot and cold feeds, if you only pumped the hot feed your water mix would be off so hot water would come through and cold water wouldnt flow into the mixer properly as its a lower pressure, potentially you could find the hot water makes its way into the cold header tank.
Secondly by not removing the shower pump when not operational poses a flow restriction so if you were to replace the shower pump (with another pump) you would want to remove the current pump to remove the flow restriction.
Please let me know if you have any further questions.
If spur off the hot water tank outlet and fit the water pump this shouldnt cause any issue to your shower pump.
Please let me know if I can be of any further assistance.