As Wendy pointed out, not having a seat for the 1 year old isn't really recommended, but it is what it is since you have already purchased your tickets. The FAA kind of shoots parents in the foot with that one: Your kid needs to be in a car seat, but wait, they can fly free under two if you don't get them a seat... hmmm... anyway, I digress...
I would say 100% bring your 4 year old's seat on board. It is the safest place for it anyway and there is no extra charge for him to use it in the seat (assuming it is FAA Approved). You can throw it in the stroller if you are bringing one (have 4 year old walk and wear the 1 year old if possible), or get a luggage cart. Or my personal favorite: strap it onto your rolling carry on.
I have heard of some airlines allowing you to use a car seat for the child if there is an empty seat on the plane, so it might be an option to bring a seat along and use it, but then you would possibly having to gate-check the seat. Which is risky, but as long as the seat is inspected and has no damage it is safe. That said, I would never check my primary seats, but a separate travel seat I would be OK with in a pinch.
I have also heard of parents who purchase a new seat and leave it in the retail packaging to check for a flight. I guess the theory is it is packaged that way for shipping and probably takes the same abuse in checked luggage. So the way home may be problematic, but if it is a backup seat, any damage from gate-checking on the return flight wouldn't be a huge issue.
If two adults are traveling, you also have the option of having one stay at the airport with the kiddos while the other takes the rental car to a local store to pick up a new seat. Same theory being that you can gate-check it on the way home and if it is damaged it won't be a huge deal. Unless you are travelling to visit someone who is meeting you at the airport, then it is even easier to just buy a seat online and have it shipped to them so they can bring it to the airport.
You can get a Scenera NEXT for $40 from Walmart that would fit the bill for any of these options and it has the bonus of being able to install easily in most cars. I am assuming that would be cheaper than renting a seat anyway.