I don’t have any experience with Lakka, so I don’t know how easy it would be to experiment with different resolutions or refresh rates, but that would be my first guess. Since it’s an HDMI adapter, there shouldn’t be any drivers per se, since the video signal in will be the same on any platform. LCD displays are sometimes more picky about sync and such, but CRTs are usually very permissive, so being slightly out of spec shouldn’t cause issues.
It’s tricky, because composite carries either 240p or 480i. 240p is out of spec for HDMI, I think, and interlaced video output is also uncommon for computers (also undesirable for gaming). 480p is generally supported by HDMI TVs, accessories etc. but doesn’t work over composite, so it’s hard to know what resolution the adapter works best with. Without any good documentation (China), you’ll probably have to experiment with resolutions to see what works, if that is the problem. Maybe try messing with the Windows settings to see if any specific changes break it (e.g. 60 Hz vs 59.9 Hz refresh, resolution, aspect ratio, etc.), then try to replicate the exact settings in Lakka. Also try a different display if you can. And test the normal HDMI signal!