"He who marches out of step hears another drum." ~ Ken Kesey
There's a big difference between removing ink and cleaning a screen. It does not take five minutes to card out the excess ink.
What I might recommend is cobbling up a cheesy device mounted to the wall with clamps to hold the screen for off-press cleaning.
We never clean on press, a waste of press time.A few years ago I went to a lecture by Greg Kitson, he spoke about using 2 carts, one for the screens going onto the press, one for the screens coming off.If we are not printing the same colors we put the squeegee and flood bar in the screen and take the screen off and put it in a cart.When we stop printing we have 3 or 4 people who quickly remove the screens and put the new screens and squeegees, flood bars in place. One person carries on with the registration and the rest have other tasks to do.I am attaching a photograph and video of the cart in use.https://www.youtube.com/watch?v=fZt4LdEVR8g