We have two dogs, which mostly get along fine. But have fought in the past. My wife just got a new job that is not WFA, so there is not always someone at home with the two of them.

The older dog stays in our bedroom, while the younger one in the living room, with the run of the house.

We also have cats, and they have litter boxes in our bathroom, so i keep the door cracked for them to. come and go. But my son's big orange cat pushes the door open enough the older dog can get out of the room.

Today when I left the house i took a piece of rope, tired it to the doorknob (door swings into the room) on the room side of the door, then using a caribeaner connected the rope to the curtain hold outside our room. This prevented the door from being pushed open any further, however when I got home the dog had pushed the door closed, preventing the cats from getting to their litter boxes.

I am looking for something cheap, or hopefully i can make, that will hold the door in place, open just enough for the cats, but not the dog.

I was looking at something like this, but we have a sweep under the door that would prevent this from working.

ETA: The rope contraption

The sweep, because without it the cats that sleep outside our bedroom at night, fight with the cats that sleep inside our bedroom at night.

My stupid cat has on multiple occasions locked herself in my bathroom by getting behind the door then flopping down, which pushed the door closed. The solution I found was "door pinch guards". Little foam things that go on the door to stop it from fully closing. Originally designed to stop toddlers from pinching their fingers in the door frame, it works great to prevent my cat from locking the door on herself. That plus your rope strategy may let you keep the door kind of but not fully open.
Something like this

This may be exactly what I needed.

Brick on floor between door and frame

What about a baby gate? Cats can jump over it as needed?

Both dogs can jump over our tall baby gate, that would be the easy solution.

I recently saw a product line called the Door Buddy that's made exactly for this situation. Here's their website: https://www.thedoorbuddy.com/

Maybe that could give you some ideas of what to make yourself (or something to save up for and maybe buy?) Good luck!

The politics between the animals in your house is insane.

This is exactly what I had in my head, but had no idea how to actually achieve it.

Edit: Never mind, it is permanently installed. [sad trombone]

We have quite a few rescues, and they all have very strong personalities.

No, because of the sweep. And I do not think door stops would be enough to hold it in place.