How to eliminate zero-thickness geomentry when using Solidify modifier and Vertex Group

I am working on a character that has fins for ears, and have set the fin membranes geometry to a vertex group, then added a Solidify modifier to the mesh to only affect the membranes vertex group.

The membranes now have thickness, but the modifier has caused rest of the mesh to zero thickness faces z-fighting all over the non-membrane geometry

You can see the effect this causes in this image (the ear fin is a separate mesh from the rest of the head).

Adding a value of 0.001 to the Factor field helps a little with the z-fighting, there are still issues where the geometry is shrinkwrapped to the head, as Z-fighting still occurs there, regardless of whether the Shrinkwrap modifier is before or after the Solidify one, and the Data Transfer modifier to blend the fins mesh with the head doesn’t overwrite the z-fighting.

I’ve tried using a Decimate modifier after the Solidify modifier to try to collapse the zero thickness geometry, but no settings I tried there helped any. Looking for any other ways to manage eliminate this problem geometry, or at least mitigate the z-fighting issue.

Goddamnit, as soon as I reach the end of my rope and go looking for help, I run smack into the answer I was looking for.

I had partially solved my problem before by setting the Solidify modifier’s Factor to a small value (0.01) to prevent most of the z-fighting. What finally cleaned up the Shrinkwrap artifacting was changing it from the “Nearest Surface Point” mode to “Project” mode with “On Surface” snapping and Culling Backfaces. Now it blends smoothly with the rest of the head.

Posting this just in case others encounter this same issue.