The previous hacks on the valve & pump continued to work.  The external battery pack proved quite a pain, so there was an attempt to make the battery contacts more assertive.

Then the end cap hole got widened so it could slide off a tube.

Then replaced the filter with a much longer 3mm x 5mm tube & 3mm fittings.

 The silicone tube is heavy enough to sink without any fudging.

This setup was bulletproof.  It could spray all body parts.  The rechargeable batteries worked.  The mane problem is it's not very portable.   The nozzle can't be folded down & it's still a bit long. 

Being able to see the water in the tube revealed a peristaltic pump would not have a high enough flow rate.  The balky suction pump fills the tube almost instantly, once it's primed.  The priming settled on sucking water in the inlet up to the water level, which is about as high as the pump can suck it without priming.  Then pinch the tube & hold the inlet upside down above the pump.

Suck in just  enough air so when the tube is submerged back in the tank, the sucked in water is below the inlet.  Then run the pump below the tank until it's primed.  It just needs  enough water below the inlet position for the siphoning action to prime the rest of the tube.  If it still won't prime, blow in the inlet to clear any surface tension blockage.  Lions wouldn't use it for insecticide.