I believe the minimum connection time is 150 minutes. And yes, as the other posters said, there are tons of GMP PUS flights.

You could also look at just KTX from Seoul Station or Incheon Airport if your timing is right. If you go to Seoul Station, that's around 50 mins to 1 hr by train. Then there are KTX trains every 20 mins to Busan that take around 2.5 hrs. Also, Busan Station is about 40 minutes closer to town by public transit than the airport.

BUT you will have to carry bags.