Buy a seat if you can afford it or the trip over an hour and a half or so. Having a lap baby, especially a super young one, is not too bad even easier if both parents are traveling and you can share the work. I am not sure about the baby bjorn seems to me that having a child in a bay bjorn is actually safer then in your arms only. Don't know the rules.
Also, be sure not to put a seat belt around you and the baby. Put the seat belt around you only ( ytou can thread it on the inside of a bjorn between you na d baby) as you weight could crush a baby strapped together.
Don't forget strollers can be gate checked and used right up to the AC door and oicked up after the flight,