Strange. I'm not familiar with the "Upgrade materials to lightweight materials" feature.
The default Shaders that Spine-Unity comes with are pretty regular PMA sprite shaders. Not much fancy going on with them. But we'll look into this process and let you know.
Note, for the most part, Spine doesn't really "care" what materials or shaders you use. We just provide you default shaders and materials so you can get things working immediately.
You can use a known working LWRP shader on all your materials. Just make sure you match the premultiply alpha settings on the atlases you export, and the "PMA vertex colors" setting on your SkeletonAnimations.
edit:
From what I can understand of the feature, it just converts PBR and standard unity shaders from the regular ones into their lightweight counterparts and settings.
That doesn't really apply to the bundled Spine shaders as they are.
Also from what I've read last, Shader Graph doesn't really support LWRP. But if it works, you should be able to use it. It doesn't matter if you are using Spine or not with that shader. It just needs to be a shader that handles sprites correctly.