You certainly need to have the relevant travel documents as if you you would be entering South Korea.

If the MU agent takes a strict view of things their ticket is to fly you to ICN, and they have to assert that you are admissable to South Korea, whether or not you do an airside transit on the same reservation to another ticket. Interlining baggage is another question, MU might check it through for you, you might be able to sort it at a transfer desk in ICN, or you might have to go landside and check back in, in which case 3 hours is very tight and offers little margin for delays.

Whilst you've managed to get two tickets in a PNR, airlines focus more on sequential tickets which form the contract rather than PNR which is a wrapper for reservations and tickets.