Here's an amateur modification of the Walter puppet beyond the basic unstuffing and mouth-insert trimming I had already done. It looked like a real Muppet to me, except for the puffy eyeballs and embroidered pupils, so I carefully removed them, leaving the lids (the lash lines came out too as they were stitched between the lids and eyeballs).
I picked up some fabric cement, a sheet of tacky-back felt, and found a couple of plastic spoons. I trimmed the spoons and glued them into the "sockets," trying to set them in a bit rather than just laying them on top of the face. Then I glued the lids down in place, and cut out two strips of lashes and two pupils that were about the same size as the embroidered ones, using a button to trace around. Luckily that stuff is really sticky, so it was easy to stick them in place, and fortunately there was enough play in the felt lash strips not to have to pre-cut them into a curve.
I didn't use the actual Walter as a model, and I realize the spoons sort of look like...spoons. But I really liked the plush when I saw it, and thought he had a genuine retro-Muppet/ventriloquist dummy vibe. He reminded me of early Sesame Street, even more than Walter himself. So I can't pretend I was making him more like "Walter," but rather taking a base figure that had a lot of potential and adapting it into something that looked like it could be a real Muppet...sort of.
One of the things that impressed me seeing the real Muppets in that museum tour a few years ago was that the parts looked very hand-made up close, especially the imperfectly round pupils. So this little guy is my own pretend Muppet, rescued from an early Sesame Street skit from my childhood. He doesn't fit over my hand, but he looks smart on a water bottle.