Have you tried:
Row.keepWithNextRow = true
for all rows in each table? [I don't think you can use everyItem() for this if your table has header and/or footer rows.]
Usually, a table appears as the single entity on a single line paragraph of its own.
Setting "Keep lines all together" for that paragraph doesn't allow the table to break as well -- even though the paragraph itself is a single line. I guess someone at Adobe's foresaw this question.
It's a slightly better solution than keeping rows together, because you can put this in a paragraph style tailored for "table placeholder only".
Thanks for looking into that. I thought that might work but didn't have the energy to check it out.
I believe that with either solution, InDesign will simply override the keep rather than go into a permanently overset state, so neither solution is guaranteed to work under all conditions.
We have tried using a paragraph style on the table and use "Keep lines all together" as I tried to express in my opening post. However that doesn't seem to work, unless we're doing something wrong with it. I have yet to had time to test out Dave's suggestion, will get back when I've tried it out.