Trains to Busan on July 4 are at 4:30 and 6:30. You simply won't have time to catch the 4:30 train with two kids, bags, etc. You'd have to hussle quickly even as a solo traveler. Also, at 3 PM, that's when a lot of flights from Europe and America start to land, so immigration starts to get busier then.

Buy the tickets online and pick them up at the airport train station. Take time to grab some food somewhere in either the airport or the train station.

I know that it may seem incovenient to have to wait at the airport, but it would be much more difficult to go to Seoul Station, transfer, carry bags up and down escalators, get tickets, etc.

If something does happen and you miss the train, they are refundable minus a very small fee, something like 5% of each ticket after departure.