I'd keep the insert with carpet. If you install a hard surface flooring into the insert, you'll still end up placing an area rug over the insert. Why waste the money.

If you want a different appearance you could always have a shag wall to wall carpet installed.

Agree with BeverlyFLA …… if you want it to look a little more like a very large area rug you could change the texture as suggested ( or a Berber or ? ) & eliminate the clipped corner, just square it up.

I don't like wall-to-wall carpet, but in your case, the most logical solution is to replace with carpet again. No other flooring type is going to look right.

The mistake was not doing this area when you changed out the rest of the tile.

Why did you keep the carpet back then, but now want it gone?

if you are set on removing the carpet, I would square up the designated living room space and remove some of the existing tile. For me, to the ceiling beam.

I'd do a mix of patterned tile with a long wood tone tile to make a pattern, I'll see if I can find a visual to post.

What is the size of the current carpet area? Could you pull up the carpet, replace with similar tile to floor and then cover the area with a nice new textured area rug so that any tile that showed would be minimal?

I think sticking with the carpeting is a good idea as well as squaring off the clipped corner. Any tile you try to put there will just look like a patch job. I'd do something with a loop instead of plush.

Wool berber is lovely underfoot, would work with the tile, and cleans wonderfully.

Thank you everyone for your comments. This is helpful.
A textured carpet seems like the most logical. We are not even sure why we went with carpet in the first place. But ten years and two kids later it’s time for an upgrade.