Flooring – Two layers of underlayment for sheet vinyl

I'd question how waterproof you need this to be. Will there routinely be water sitting on the floor? If not, you may be over-engineering things.

That said, off the top of my head, one option for waterproofing the subfloor would be to paint on Redgard:

It's a 'paint on' waterproofing membrane that you'd typically use behind tiles on cement board and the like.

Vinyl plank will allow the urine to seep underneath, unless you use a sealant on the joints/seams. Even if it is properly installed.

Using sheet vinyl will work perfect for your application. The only way urine or any other liquid could get underneath would be if a hole/tear was made into the vinyl from mistreatment or from a big spill seeping in from under the walls.

Also you can get the room floored cheaper. You can also get some sheet vinyl with wood grain in any tone/shade to match the room decor. If you do need a seam in the room a seam sealer is used - like epoxy, so no leaks

I've been installing for a long time and I have dogs that have accidents. I have sheet vinyl down in their room (12'x18'). No leak throughs. Easy cleanup, no joint lines to scrub.