It may be possible by pushing in your own time variable using a global float value. You can right-click a value property node and make it global, then push values into it using Shader.SetGlobalFloat(...)

I've planned to clean up the entire lighting pipeline pretty soon, which, as a side effect, there will be nodes for both lightmap data as well as light probe data

Cool, thanks for checking :)

I'll have a look at it tomorrow, thanks for the feedback

Does enabling probes and/or ambient light help?
If not, does plugging in something into spec help?

Only in deferred rendering I presume?

This should work - how does your node tree and settings look?

Is receive shadows checked on the object itself?

Under Syntax, there's a description of offset:
https://docs.unity3d.com/Documentation/Components/SL-CullAndDepth.html