Your family would pay for the flight for all four of you? Fly, for sure.
Think of this: You can tell your mom what bottles etc. to purchase and have ready down there; you don't need to carry many toys, or tell mom you'll ship some. As for car seats -- the kids should be flyiing IN those, strapped into their airline seats, anyway. Call the airline and say you will need priority boarding and help with three car seats (or two, if the five-year-old is large enough to fly without one) both getting on and off the plane.
Do not have a "lap baby" or you will be exhausted by holding a wiggly kid. Get each child a seat on the plane and use the car seats in them. It will indeed seem like a hassle for the time you are in the airports, but it will be over much faster than if you drive and must listen to crying or "I'm hungry" for hours.
There is a world of difference between going all around town all day in the car and doing a 13-hour drive with three small kids. Plus, statistically, you and they are safer flying than driving. If you do not have another adult to share the driving -- you will be driving tired, with no one to help out. Dangerous. And we just returned from a long road trip, and other drivers, especially truckers, drove like maniacs on the highways. Do you want 13 hours of that, alone?
Reconsider all the stuff you think you need for two weeks. Other than car seats, take only the most basic clothing etc. You can wash while you're there-- no need for a lot of clothes. Even consider shipping clothes ahead etc. Strip down you list of "musts" and tell your mom what you need her to get or what you will send on ahead.