When I used to have cages I cut pieces of plywood to fit into the gap. They were only thick enough to fill the gap, meaning the door would still slide shut.
They were long enough to only fill the spot between the other sliding door (the one that lays closer to the plastic part) and the side of the cage.
Of course that kept the other door from sliding , but all you had to do was open the side with the wood first and move it out, and you can move both doors if you have to.
Easy and quick and most of all effective fix. Found an old picture that kind of shows what I mean...
Hope this helps...
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