The little voice in your head whispering: “Get rid of it,” is just hormones. Ignore it and it will go away. The newborn stage is stressful but it passes. I think you would regret bundling out your pet down the line.
Let the cat go outside. The poor thing is begging for attention because it is bored and under-stimulated. Yes, there is a certain amount of risk in letting them out but they love it. And it is better than being sat in a shelter competing for the chance of a home with this season’s batch of incredibly cute kittens.
Get a robot hoover, a net for the crib/cot, a brush for DH to use on the cat and close the door of the baby’s room.
I have had two DC and each time I had cats. The cats, wisely, tried to give the new noisy things a wide berth in the early days but as time went on it was lovely to see how relationships developed between them. I have very fond memories of one lovely, tubby, placid girl in particular, who allowed herself to be treated as a living teddy bear by a toddler.
Both my DC grew up to adore cats, and the cats were, and are, an unending source of comfort and companionship for them. Heartbreak, exam stress, friendship breakdowns, family squabbles and so on - there have always been one or more friendly, cuddly cats happy to see them and snuggle up to them.
Don’t make a decision you will regret.