If you're on a "Light" fare you do not get a free checked bag:
https://www.swiss.com/us/EN/prepare/...hecked-baggage.
The K fare is one of the lowest buckets so most likely it fits in that category. Do you have any status with UA or another *A program? That may solve the bag fee issue.
On the plus side, at least K is earning on LX; it would get you nothing on LH.