Realistically, Ludwig should realisitcally be AP stomping here considering the fact he scales relative to weapons like the Kirkhammer and what is basically a downgraded version of his own sword, both of which are 8-A+. Granted those calcs are a bit sus, but they are accepted and just not listed on his profile.
Anyways, yeah might have to give it to Ludwig. They have pretty comparable attacks, with both of them basically firing off energy beams. Ludwig also just has better LS so Homelander has no chance to really box with him, and better survivability with immortality stuff. I imagine Homelander getting a few shots in, thinking he's killed Ludwig, before coming in and getting bitched or caught off guard considering he's never had to deal with the level of Immortality Ludwig has.