The train works well. I've taken the KTX from Seoul Station and the SRT from Suseo Station. I found the SRT to be a little more comfortable - more leg room, for one - but Suseo Station is not centrally located (I had meetings not far from it, so the SRT worked better than KTX for that trip).

I've also flown Air Busan a couple of times, though maybe your travel office also objects to them. Cheap, easy, and got me there on time. You'd need to take the train from ICN to Gimpo, but that's easy, and if you need to end up in the Gimhae area when you get to Busan this may be the better option.