To answer the question in the title: I have all the novels listed in the Heinlein bibliography on Wikipedia, and too many short stories and anthologies for me to want to check them against the bibliography. In none of them have I found anything about dealing with a cat's bowel motions in zero-G.
The only novels of Heinlein's I've found that mention a cat other than in passing are:
For Us The Living
Space Family Stone
The Cat Who Walks Through Walls
The Door Into Summer
The Puppet Masters
To Sail Beyond The Sunset
and none of these mention the subject (incidentally the cat in Space Family Stone is an alien flat cat). In fact only Space Family Stone and The Cat Who Walks Through Walls deal with weightlessness at all.
The only short story I've found is Ordeal in Space.
I'm reluctant to claim definitively that Heinlein never mentioned a zero-G cat tray (there may be other stuff of his I'm unaware of) so I'll content myself with saying only that I have been unable to find such a mention. The nearest Heinlein comes is in Space Family Stone. The family are loading up the bicycles they're going to recondition and sell on Mars, and there is the following exchange:
‘Mind you don’t try to pass them off as new. But it looks to me as if you had taken too big a bite. When we get these inside and clamped down, there won’t be room enough in the hold to swing a cat’ much less do repair work. If you were thinking of monopolising the living space, consider it vetoed.’
‘Why would anyone want to swing a cat?’ asked Meade. ‘The cat wouldn’t like it. Speaking of that, why don’t we take a cat?’
‘No cats,’ her father replied. ‘I traveled with a cat once and I was in executive charge of its sand box. No cats.’