We have 4 crib loving cats. We went and bought an inexpensive, wooden framed screen door (seriously is just a thin wood frame and screen) and put in up in the doorway to the bedroom. We didn't put any of the hardware on except the simple handle, so Owen can push it open if he shuts it. Its only shut at night and nap. It has worked really great for us. As you can see we didn't have to remove the regular door.
Anyhow, it's not a solution for everyone cause it does kinda look funny lol. Frankly I don't care how it looks... I didn't even stain the wood lol - it serves its purpose and actually we will be taking it off soon cause we shut his real door now when he sleeps. I got a "Crib tent" but it didn't fit as I have one of the new styled conversion cribs, so DH came up with this idea! HTH Good Luck.