My kids are almost grown now, but I *always* booked a separate seat for each one, and I *always* used a child safety seat in flight for the same reason I do in a car: my child's safety. If you can afford a third seat, booking it will be safest for your baby and IMHO make a flight less stressful, too. My kids were familiar with--and comfortable in--their usual seat from the car at home, and I knew I had a clean, safe place to set him down when I needed my hands free.

BTW Checking in a child safety seat as checked luggage is a bad idea, too. If the seat is dropped by careless baggage handlers, you wouldn't necessarily know, and a hairline crack would mean the kid was not safe in the car thereafter. That's another reason to use the seat on the plane instead of opting for a lap infant (unless you have a safety seat waiting on the other end, like at a relative's house.) Since automobile travel is far less safe than flying yet done much more frequently by most Americans, this risk is even worse than the in flight safety risk IMO.

Getting through the airport can be made easier with a wheeled cart for the car seat. I had a special car seat trolley that attached with two bolts to my Britax seat and stored in minimal space, but I flew multiple times every year as a solo adult with two kids a few years apart, so it was well worth the investment for me back then. Travel as a couple is MUCH easier. Your own stroller, an airport luggage cart, or a suitcase trolley could work to reduce the effort of dragging a car seat around. If you aren't transporting baby strapped in the car seat (and I had to when alone), it doesn't have to be as secure an arrangement.

I never once had a hysterically screaming baby for the short period of takeoff/landing when the child was required to be strapped in. This could be an issue if your child is more sensitive or has colic. If you use bottles at all (mine were primarily breastfed, but neither objected to a supplemental bottle), having one ready was sufficient to keep my very young babies calm during this stretch; pacifiers worked for one of mine as well.