I think it will depend on what tile size/shape you pick. You won't want stools on or sliding the carpet, so maybe something like 3' past the bar? That 5th stool on the end will be tricky though... Just looking for 5 seconds, maybe don't extend the tile to the left of the bar and leave that be standing space?

Honestly if this was my basement, I'd do LVP for most of it (would get rid of this tile/carpet issue) and only put carpet in the bedrooms, then maybe an area rug under the couches. Then you could rearrange or re purpose areas and not be married to anything. I do like carpet for tv areas or if kids will be on the floor, but this open concept makes it difficult

LVP (luxury vinyl plank) and LVT (luxury vinyl tile) are pretty good looking and very thick. I have both in my basement right now. They are thick enough where you don't really feel the coldness of the concrete. My basement bathroom is the LVT, and it's thick enough where it actually gets grouted so that looks pretty real. It's also warmer on your feet than tile is

I don't know any brand names, but most flooring stores carry both if you ask. The stuff is extremely popular, and many people on this board intentionally do their entire houses in it instead of real hardwood/tile

Place tile where it has a potential of getting wet.