It has been an agonising three weeks. Your mind has never been this preoccupied. If you were younger and more reckless, without a job, you would have walked away from the problem. But now, you are old, you have morals, and this has to be handled in the most mature way. You will give Carol two options: marriage or child support. Yes you are dead serious that you want to list the kid as one of your heirs.
They are the most neutral choices you can give her. The choice is hers. You have already made up your mind. You have figured out that in the event she wants to move in with you and start life, it won’t be a problem. She will just move to your apartment in the corner next door. You will see what things she will bring over, and what you can sell.