We fly Southwest all the time and are a family of four.
When we flew with ds under 2yo we also purchased a seat for him and used his car seat.
You only need his birth certificate if he will be a lap child...since he is a ticketed passenger you won't need it.
What I do suggest is that you OR your dh board first w/ the car seat and your older child. Get 2 seats each in 2 rows in front of each other. One parent sitting with each child in each row. If your little one is still rear facing, I would put your older child in front of the younger one (car seat must be installed next to the window). The older child will not be as upset about not being able to recline as another random passenger!
I love keeping my kids rear facing on the plane...they are facing me, easier to keep them occupied and...less stuff to fall on the floor!
Have the parent on the plane install the car seat and settle as many carry ons as they (and the 9yo) can carry on w/ them. The other parent keeps the younger child on the go in the gate area until late in the boarding process. Less time on the plane and the extra activity may tire them out so they nap during the flight. (This worked really well for my dd, not as well for my ds!).
Another thing that worked well on the plane w/ the kids (ours are 4.5yrs apart) is to give each kid their own backpack filled ONLY with toys/books/other amusements. Any snacks or diaper bag items go into mom or dad's bags (we have an excellent backpack diaper bag from Land's End that one of us carries). Our older one has her DS, a book or two, activity book, card game, and some small toys (currently Littlest Pet Shop), the younger one at under 2 had a couple of board books, some Fisher Price Little People, teething toys, etc.
Good luck!