We're having a similar issue where one of our characters' layers are rendering out of order on LWRP – specifically, a character with more than one material. I did notice that LWRP does not support multi-pass shaders, which the Spine/Skeleton shader does use. However, commenting out the later pass (which I think is for rendering shadows, which we don't need) does not solve the layering issue, so that might be a red herring.
As you say, does seem like a LWRP problem with multiple materials, but I haven't been able to pry out any further information about that.