I was really interested to see what that layout looked like underneath. I guess the unclosed table explains it. Still interesting, as the HTML is injected so it shouldn't really "leak" per se. Oh well. Glad you figured that out.
Seems you've figured this one out, too, but there really isn't a way to force a cell to use a different property set on a different screen. So if you've got a cell used on 2 screens, it's going to use the same property set at any given time during a tournament. To accomplish what you're asking, I think you should make a copy of the cell and create a different property set with different conditions.
As you noted, "copy" doesn't copy conditions. But that's not entirely true. If you copy a property set from one cell to another, it's only copying the properties themselves - not the conditions. What you're probably wanting to do is copy the entire cell (including property sets with their conditions), then make changes to the new cell. To do that, right-click a cell in the Toolbox, and select Make a copy of <cell name>.