My first thought would be a T connector attached to the water outlet at the place where the hose attaches to the pipe coming out the wall. On the third leg of the tee add a valve, and then whatever bell reducers you will need to adapt to the washer.
By putting a valve on the leg going to the washer you would be able to turn the water off on that side and still run water to the showerhead.
By the same method, switching to the valve your suggesting, you could put the T and a valve on one leg right there at the faucet.
Both of these suppose that you have a valve on the showerhead itself to stop the water from flowing through it.
But compared to either, it would be easier to just remove the handset from the end of the hose and connect that directly to the washer. But you will have to connect and disconnect for each usage.
I bought something like this years ago for washing my dog, could you adapt it?
Sink adaptor
