The problem is the pump is pulling air in from the open vent.
The simple answer is that you should not fit any valves to the open vent; you should increase the available pressure at open vent tee by;
a) raising the tank or
b) reducing the pressure losses due to friction along the flow path from tank, to cylinder cold feed to cylinder outlet, etc., and/or
c) reducing the pressure losses due to friction by reducing the flow rate by throttling the pump's discharge (or not turning the shower on full-blast).
Regarding b above, there are pressure losses due to friction along all the pipe and all the fittings that the water flows through. You can reduce the pressure losses by using larger pipe, using pulled bends rather than elbows,using full bore ball valves (rather than cheap service ball valves or gate valves), using copper rather than plastic (larger bore, no inserts at joints), etc.. Any restriction causes pressure losses. You can find tables for these pressure losses at a given flow rate and can calculate the pressure at any point in the pipe.
The pressure at the open vent tee is equal to the static pressure/head MINUS the frictional pressure losses. In your case, the frictional pressure losses are too great and/or the head is too small; the pressure at the open vent tee is less than atmospheric, and air enters the pipework.
You could try running the shower and gradually increasing the flow rate to see at what point air is sucked in. This would give you an idea of how severe the problem is and whether it could be fixed by replacing a few fittings or whether it need a whole rethink.
PS Fitting a non-return valve in the open vent would stop air being sucked in; you'd also get cavitation in the pump which would wreck the plastic impeller.