1. I could see if the pump is actually pumping fluid i.e. does fluid come out of the cut end.
2. I could try the bicycle pump idea and see if I can force water manually through to the tailgate end?
When I did it I removed the valve. The open end of the tube coming from the reservoir sat conveniently in a sort of gutter and when I operated the washer switch water and black gunge could be seen in the gutter.
I connected a bicycle pump to the tube leading to the jet via an adaptor for blowing up Lilos etc and forced air and some water through the jet.
I was able to poke some gunge out of the valve with a piece of wire and left it soaking in sterilizing solution.
The bicycle pump was actually a track pump with a connection like on a footpump with a camming lever which compressed an O ring that fitted round the valve and let me blow water and air out of that
Unfortunately you would appear to have a blockage or leak closer to the reservoir.
What I would do is take a pipe off at the tailgate and SUCK to see if you can pull anything through. Just be prepared to SPIT and have a peppermint handy!
Having seen the gunge that came out of mine I would advise persisting with the pump.
You could perhaps suck with some sort of syringe arrangement.
Or if you can find one of the joints closer to the reservoir to access and blow towards the jet with a cycle pump or compressor. No need to cut anything.