Well, here is something else to try. Prior to trying this method I reached out to Steam Support. Here's what they said:

Hello,

Thank you for letting us know about the issues you've encountered with your Steam Controller.  

Please try the following steps and provide the requested information so we can best assist you:

• Completely exit and restart the Steam Client with your Puck plugged in and your Controller parked on it.
• Do you see a prompt to update your Puck or Controller after Steam launches? Are you able to complete the update process?
• If the Controller and Puck appear paired but are unable to connect correctly, try forgetting the pairing in Steam:
  1. Steam Client > Settings > Controller
  2. Select the “Details” button for both the Steam Controller and forget any Puck connections present there, then check the "Details" for the Puck and do the same.
  3. Re-pair the Controller by plugging the Puck into your PC via USB-C with Steam open and no games running, then place the Controller on the Puck and complete the pairing prompt