Replace inlet screen with washer screen on washing machine?
I'm not familiar with cleaning these, just installing new stufff.
If this filter behaves as a washer does, I'm suggesting that you install another one at the valve end of the ( stainless mesh wrapped) flexible supply hose. When it gets embedded in the female threads, just dip these ends in the cleaner without removing filter screen.
I've had clients whose water supply was switched from middle of a zone to bottom of a run of hillside water main ( Berkeley, EBMUD) and had to clean the East Bay Mus from every screen in every faucet every month since.  

I'm about to receive a new water main. I'll have to turn off toilets and blast outdoor spigots for a time until chunks are flushed.

Some toilet float valves have a twist drill shaped filter that catches lots of chunks but never fully clog up.  I wish there was a better angle stop ( valve) to turn off toilet for cleaning, those darn things leak from gunk stuck and I have to disconnect and discharge into buckets.

Too wordy? RSVP. Thx
I'm mostly a carpenter, plumber only in emergency