Oh Miroslav

The fact that you're caring is your biggest possible pitfall here.
I'm sad to learn that heartache has been such a big part of your life, but I hope that means you've had a lot of loving too, as the counter point.

To not have had a break away longer than 2 days in 20 years is a huge commitment that you've given to others, so maybe now is the time to be a little selfish for once, and think of yourself?
Would it be worthwhile suggesting to the people who've asked you to take this cat to consider working with a pet behaviourist to see if they can establish what the problem is with the cat that's weeing (whichever one it turns out to be?) It seems to me they need to show a little commitment too. If they dont want to (or cant afford to) what about them reading a few useful books or articles about cat behaviour?
The second article on this link may be useful for example
http://www.wayofcats.com/blog/In case the link doesnt work, this is Pamela Merritt's reply to a reader who had a question about cats who wouldnt use the iltter box after an incident.
"If our cat has an incident with the litter box, whether physical (pain and illness) or mental (being startled in the area,) they could be practicing avoidance even after the incident has been taken care of. We need to get them to see the litter box as their friend, again.
Here’s the steps to reboot our cat’s head:
Illustrate the issue. We need to let our cat know we understand that something had gone wrong. We’ve been to the vet, or fixed the rattling blinds, or cured the cat conflict that had them being jumped on their way to the litter.
Open the subject for discussion by visiting the litter and asking thing how things have been going, lately.
Demonstrate the solution. Show them that we’ve finished the antibiotics — Look, I’m throwing the pill bottle away! Mention that their friendship a former rival has improved — Mr. Whiskers isn’t being mean to you anymore!
Whatever the problem, you have taken steps.
“Fix” the litter box. We know the problem wasn’t with the litter box… unless it was. In any case, we have to change something about the litter box, so it is now a New Litter Setup. Choose a different, but similar, litter. Get a new box. Move it a bit.
This is tough, because we often feel that we had chosen the best setup already! But remember, once we have them using the box again, it’s easier to change it back.
Encourage them to visit. Get excited and lead them over to the litter box. If they don’t mind being carried there, try that. Call their name, and see if we can at least get them to stick their head around the corner.
Don’t despair if we don’t get much cooperation at first. We have piqued their interest, and they often investigate on their own after we’ve alerted them.
Create new routines. Make a big fuss over cleaning the box. If we were already fussy, go to the box and exclaim over how clean it is.
Make a point of cheerfully visiting the box, with a big show of inspecting everything. It’s because we care.
Remove anxiety. If our cat is still apprehensive, we need to address that. Try the tips in my post about reducing cat stress. They need to calm down so they can think, and realize things are better.
Now we and the cat can enjoy our new, and improved, litter box."
To work with the cats in situ would seem to be the ideal solution, as the cats wouldnt then need to be separated. They could try the litterbox in a different room - the bathroom or kitchen for example.
I appreciate this isnt really your problem, even thought they may be trying to make it your problem, but I hope they would be willing to invest some time and effort into sorting this out, for their benefit and ultimately for their cats. And bless you for caring about the outcome.
