Try powdered gelatin. Using these two recipes as reference (1, 2) I recommend you start with a tablespoon.
In candy making, gelatin is a very common way to create chewy textures, like in gummy candy or marshmallow. It turns out it can do just the same in baking. While candy recipes generally require you to "bloom" the gelatin powder to avoid clumping, this can be avoided in our case: you can simply whisk the powdered gelatin in with the rest of your dry ingredients.
Gelatin is a protein, so it definitely meets your zero carb requirement. It is also inexpensive when purchased in bulk (the tubs are usually cheaper than the packets).