Shower to Washer Project - Recommend an Adapter

THIS should work for you. Be sure to get a roll of teflon tape if you don't have one....Dennis

More importantly, won’t it be kinda cramped taking a shower now? Drying off might be easier though; put it on spin?

:):)

Will the washer's feet or movement damage the floor of the shower from it's shaking.

The washer should be hose thread not pipe thread.
Unscrew shower head and get a 1/2 inch female x hose adapter and attach to washer with a washer hose.
Looked at your link after I posted. That top thread looks like it could be ips(iron pipe size) so should still work.
Link to your faucet HERE

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