I would hinge the door and treat it like another cabinet.

Or put the trash pullouts back in and use them for dog food etc. A door on a pullout makes no sense.

I have 2 small trash cans under my sink I use one for real trash and one for recycling and anoter bin for compost so not sure why you do not needtrash cans for at least those reasons. The door on the pullout to allow access to the bins is always how it needs to be IMO I like the dog food idea I have ine in a very expensive stainless step can I have huge dogs and a small trash can would not work.If you want to check out rev-a-shelf you might find something to fit there for some other purpose . Where is your trash now ?

We recently bought a house built in 1995 that had mostly base cabinets. I added rev-a-shelf pullouts to most of them (including a trash/recycle). The door attaches to the lower pullout, so it functions as a drawer within a drawer. Not as good as actual drawers, but a ton better than base cabinets.