Edit Mode: "Mesh Symmetry"
Rather than using Blender's "Mirror Modifier", you can opt to use blender's built-in "Mesh Symmetry" modes, which can be enabled by selecting your "Symmetry Axis" to the right of the butterfly icon.
This will not work for non-symmetrical meshes, and cannot be used to create new geometry (i.e., you cannot extrude with symmetry, but you can grab with symmetry.
I recommend that you create mirrored base-mesh, via the mirror modifier. Once your basemesh is complete, apply the mirror modifier, and dissolve the center edge loop. Finally, if you find it necessary, you can tweak your mesh using the "Mesh Symmetry" options.
Let's try this together:
Let's make a cube, deleting half of it so that we can add a mirror modifier to it.
Let's create edge loops surrounding the face, to prepare for the eyes, ears, and mouth.
Inset the selected faces to create the eyes, ears, and mouth. (Join the mouth to the centerline, for a proper face. Here I have removed the geometry which was skewed from the inset.)
Add as much geometry as you think you will need, and apply your mirror modifier! (You must be in object mode to apply your mirror modifier.) Return to edit mode, and change your geometry using the "Mesh Symmetry" settings!
Here was my end result, which looks more like a helmet than a head:
In the end, I find this to be a total pain! I would recommend low-poly retopology over a sculpted mesh, or traditional base-mesh modeling. However, if it works, it works! Let me know how I can help you.
Good luck! Happy blending!