Now I have been told by a Korean that the bus is more convenient than the train because it is closer to the arrival hall and easier to manage with luggage.
I prefer trains over buses because it is easier to get on the right one going to the right place and you can stand up and stretch, but not if it adds tremendous hassle.

I know that sometimes locals haven't used the transport and are just reporting what they have heard. Anyone used the bus between the airports in Seoul?

I have used the bus between airports....but it was going the other way...from GMP to ICN. The great thing about this direction is that you can check your bags in at your originating airport (for me it was Gwangju) and they check your bags all the way through to your final destination. If you do this, you have to check in at a special counter at ICN. They confirm that your bags did make the transfer. It was easy and allowed me to bypass the huge lines that can form at ICN.

But.....going the other direction from ICN to GMP --I am told is not that convenient because as you stated, you have to claim your baggage and go through customs and then haul it yourself to GMP. Because of this, I opt to take an express bus (4 hours) from ICN to Gwangju. All I have to do is haul my bags out the door at ICN to the bus stop. It is pretty easy and since I am usually trashed, I just sleep all the way on the bus.

But....going to CJN requires a flight....so yeah, you are going to have to transfer, as you state in your opening post. I talked to a Korean that was visiting me here a few weeks ago and she said to take the bus because it was easier with luggage. We didn't get into the details, but I wanted to pass on her recommendation.

Thanks, that helps.
The GMP - CJU flight will be my third of the trip, so I want to keep the transfer as simple as possible.