OZ (Asiana) isn't a budget airline BTW. They're KE's primary competitor and in Star Alliance. KE is trying to take them over in a merger, but has hit quite a few regulatory hurdles.

Most of the budget carriers to Korea go from NRT, and even the ones at HND will go to ICN, not GMP. But having said that, budget carriers in this part of Asia are more hybrid, so might be worth a look and compare prices. Peach out of HND is more of a true LCC, but something like JinAir or AirSeoul from NRT are run by KE and OZ respectively and very much hybrid models, not nearly as painful as a normal LCCs.

There's exactly zero chance I'd pay JL $900 for economy on that route. I think even KE is pretty high. OZ is in the range of reason, but like I mentioned, JinAir and AirSeoul are quite good for a less than 2 hour flight. That's a good route to use Avios on as well, JL has business available quite often for partner awards.