cats are sneeky. can suffercate a child in a moment. i would go the screan door. the holes can be patched up and re painted later. better safe than sorry.
As a cat owner who has cats that have not shown any attempt to sleep with either baby, I would love to know of any actual cases of cat suffocation. I find whilst my cats love to sleep in any bed, once they work out a baby is there, they jump out very quickly all without any training from us. I have heard to be careful with cats, but honestly have never come across any news article or verifiable incident where a baby has died due to a cat.