I pray to the star trek god's every night that there is no David Marcus in the next movie.
In wishing for the next movie to show Kirk's continued character arc, I would submit that highlighting his personal life is so not the way to go.
This is about Kirk and his relationship with star fleet, his crew, and the universe at large. It is not about Kirk's personal life. His growth, his ability to command would be based on the premise that he has learned some hard lessons about humility, arrogance, etc. One of those lessons would be do not get involved in a personal relationship with a crew member. Do what is right and work on himself and personal growth.
To be involved with Carol and then having David would just be a distraction and the same old, same old. He needs to have complete focus on the task at hand which is to be Captain Kirk.
Perhaps the way to go would be that he has an actual mature friendship with a female and shows he has learned the value of people including woman and that he doesn't need to sleep with every one he has ever met.
Learning to control impulses is the sign of a good captain. Yes TOS Kirk had his fun but it never interfered with his running of his ship.
Basically IMO it would just plain suck if there was a Jim/Carol thing. Not to mention if wouldn't be a good idea for a Bones/Carol thing for the same reasons.
I will scream bloody murder if we end up with Carol having David in another Kelvin like moment.