I think more photos of all the walls will get you the help you need.

First off, is this dilemma just for the holiday season? The rug is too small for the space

It's just for the holidays. I was thinking the same thing about the rug, but any bigger and it cuts into the area between the living room and the kitchen which makes it look even more awkward. The spot where I was standing is the entrance to the room

Without a floor plan, I can't tell exactly how much room there is on the stretch of wood floor that's opposite the fireplace (in your pictures, it's left of the rug and ottoman--maybe it's between the kitchen and living room?) but could you move the couch that's currently against the windows so that it faces the fireplace? Rotating the sofa 90 degrees would give you more space in front of the tree and make a nice cozy spot facing the fire. It's a low enough sofa that it wouldn't block your sight lines from the adjoining room to the fireplace, and there's no reason the furniture has to be against the walls. It would also open up more of your window view.

This is the floor plan. The couches we have is the Maliyah from Structube. They're quite large, so if we move them to face the fireplace, the space feels tighter!

Put the tree somewhere else. It looks too squished in that corner, anyway.

Rug doesn't fit regardless of how you arranged the furniture.

You say you had to put the tree in the corner, but could it be moved to other side of the window, so it's on the imaginary line between the kitchen and living room? If there's no electrical outlet there, an extension cord run behind the sofa would be pretty unobtrusive. Moving the tree would allow you to put your sofa back where you had it before, and enjoy the tree without having to squeeze between the sofa and ottoman.

Unfortunately, with an almost 2 1/2 year old, a very mobile 7 month old, and our dog, that tree is a recipe for disaster! We'll see though... just trying to convince my husband that moving the tree will be a good idea.

I'd put the tree on the other side of the couch on the short wall by the window and scooch the couch up to the wall. You'd get to see the tree from the entry that way. You could put a little baby play fence around the tree while the kids are up and remove it for adult time at night.

With 7 pieces of furniture to work with, consider storing something wrapped in plastic in your garage for the holidays.