Not with keyframes or drivers
The overlay is a member of the 3d view space.
bpy.types.SpaceView3D Each property in blender has an
is_animatable property, declaring if it can be animated. Visually if you right click and there is no add driver or insert keyframes
Step thru this in the python console where
C = bpy.context
Find which area is the 3d view
>>> for i, a in enumerate(C.screen.areas): ... i, a.type ... (0, 'PROPERTIES') (1, 'CONSOLE') (2, 'VIEW_3D') (3, 'TEXT_EDITOR') >>> a = C.screen.areas
Its 3d view space will be the active one
>>> s = a.spaces.active
And here is the overlay wireframe threshold
>>> s.overlay.wireframe_threshold 0.0
Let's see if the property def says it is animatable.
>>> p = s.overlay.bl_rna.properties['wireframe_threshold'] >>> p.is_animatable False
Confirmed ... Nope.
There are some dodgy ways this could be achieved, but will wait for further details on animation required to elaborate.