Agrees, this^^^ might be a good solution.
If it's even a problem that the cat is sleeping there.
How many birds and how many nests do you have?
Are the birds laying elsewhere....do they really want the 'cat nest'??
A 'coop cat' that doesn't hurt chickens might not a bad thing to have, can keep rodents down.
I was thinking the same thing, that a coop cat would be a good thing. I have 3 barn cats and they all leave the chickens alone. When they were younger they tried stalking the chickens, but one of the hens had the good sense to charge the cats and peck them on the top of the head. One of the cats did a back flip when it happened. Ever since that incident, the cats no longer bother the chickens and ducks!