I put it aside for a while, and finally decided the rest of the clock was just too great to not have on display. So I had another project on my hands.
Sorry I don't have a complete set of photos to document what I did, but I'll explain the steps.
First I carefully removed the hands and the movement. All of the pieces got bagged and labeled. I used a very small flat head screwdriver to carefully pry up the numbers and other pieces from the face. Each piece had two small plastic prongs on the back which go into corresponding holes in the face. That work left me with just the blank, tape-damaged face you see here.
After that all dried, I just reassembled the clock. I did have to use a small pointed pick to open up the tiny holes which held the prongs behind the numbers and other pieces that went around the clock's face.
So here's the final product, now hanging above my office door at work.
I'm pretty happy with it, but I can see that faux wood graining like anything else requires lots of practice and experimentation.