I think there are a couple of things to consider.

If the door is in the dining room, will you lose a part of a length of wall that would be the insightful place for a sideboard or other piece of furniture.

If the door is in the living room, will you throw off the symmetry of the room (you could add a built in on the other side of the fireplace to correct this.)

If the door is in the living room, you will have some degree of a sight line into the room to supervise when you are in the living room, and the kitchen, but not when in the dining room. As you probably spend more time in the kitchen proper and the living room, this may prove more useful.

You already have a doorway to the porch in the dining room. Adding another in the DR may make the dining room feel like a passageway.

Or one thought you could also could also consider would be doors in both DR and LR, thus allowing for perfect circulation on the floor. French doors would be nice for this, and that will also allow additional light into the new addition.

I also might add that even though you have heard it many times, kids grow really fast. Size/scale the space for 4 adults, not 2 adults and 2 kids. By the time kids are in 4th grade, they physically take up as much "space" as an adult

i think they're should be a door to the screened in porch if at all possible. and i'd make the new room the dining room, unless the kids are old enough so you don't have top keep an eye on them still.