A thumbnail seemed to be the best tool for this job. The casing isn't glued. There's just a friction-fit post that goes into a receiving hole in the cover. Work around the cover edge and the base part will begin to come loose. Don't get into a hurry because the wires connecting to the LED module inside are very fine and easy to break. We'll be re-using these wires, so be careful.
Step 2: Remove the LED module
The LED, connected to its base, is friction-fit into the base of the plastic candle flame. Twist slightly and pull to remove. Make note of the wire colors, as they may be different from the unit I used. I'll be using "yellow" for the negative and "red" for the positive.
Step 3: Move the cathode wire
We won't be using the original circuit, which is a low-side switch that just flickers the LED off for a few milliseconds periodically. Carefully unsolder the yellow wire and move it to the LED cathode on the middle pin. The wire is really fine. Use a hot soldering iron to melt the original connection. Add a little fresh solder to the center pin. Then you can hold the wire against the center pin and reflow the solder joint easily.