I would for sure take the car seat. You will need it if you get in a car and it keeps them contained. And more familiar with the space. You can get attachment options the connect it to your rolling suit case or even turn it into a stroller essentially.
I just flew solo for the first time with my 2 year old last week and the car seat was a life saver for keeping her strapped in. I kept it on a dolly bringing it through my he airport and essentially used it as a stroller by having my kid sit in it while we got through to the gate. as I didn’t bring one with me this trip. I asked the gate agent if I could go on as early as possible so I don’t bump people with the car seat (since depending on the plane you’ll have to hold it above the arm rests as you go through the aisle). Of course get as many wiggles out before getting on the plane as possible. I always like to bring a kids yeti/hydro flask thermos with me and stop at a cafe after tsa for some warm whole milk for take off so she’s drinking or snacking and her ears pop. When boarding, as I was struggling to get it down the aisle the flight attendants instantly offered to carry it down to my seat (Alaska airlines). So that’s as super helpful.
@Kate the friend picking us up/who were staying with has an extra seat so she will be safe in a car either way!
I suggest trying to stay relaxed and enjoy it! Know you are super capable and she’s probably going to love the experience! I started both of my boys super young 6 months and 4 months but when they got a little older I just try to keep their attention on the next task, and sit them by the window to redirect their attention if they get restless. Most of the time they just sleep though! We used the car seat once but didn’t like it because like you said super bulky and if they get fussy you take them out and then you have even less space. Good luck, it’s gonna be fun!