The neighbour tells me it tested at 12.7 while in gear. He’s not going to do anything more for me, it appears.
In gear? The Ready indicator illuminates when the key fob is present, the brake pedal is depressed and the power button is pressed. It’s what we all do prior to driving the car. I trust the only gear that the car was in while your neighbor was under the hood measuring the battery voltage was Park, with the parking brake set.
If that is the case, the 12.7V reading suggests that the DC/DC converter is not functioning properly. It could be the converter, or it could be a failed or damaged component in the circuitry that is preventing a charging voltage from reaching the battery, such as a relay, fuse, wire or loose connection. I do not have access to schematics for the Clarity Power System, and even if I did, I would be hesitant to offer any guidance, given that you do not even own a multimeter.
Now, to address your questions.
I believe the DC/DC converter, and/or associated circuitry are likely culprits for causing the Power System indicator to illuminate. The 3 main reasons I’ve reached that conclusion are, the 12.7V reading, the fact that the car has powered on and off multiple times with the same battery and that you have only reported a Power System indicator on the dash. A failed/failing 12V battery typically manifests by displaying multiple warning indicators.
Tell the Honda dealer what you know. What you should expect is for them to tell you to bring it in for a diagnosis. Ask if they will charge a diagnostic fee and if so, will any or all of that fee be applied to needed repairs. Alternatively, you could take the car to an auto parts store that offers free OBD code reading to see if the code provides any more information than a “Power System”problem. They may also be able to clear the code. If it doesn’t occur again, then it may have been a false alarm or it could be an intermittent problem. That service is readily available in the US, not sure about Canada.
You could put in the new battery, but unless you’ve had the old battery tested, you don’t know that it is near the end of its life. Bear in mind that the battery in the Clarity is not used to start the engine. Replacing it won’t help if the battery isn’t the problem. It’s just throwing money at parts without a diagnosis. However, if the converter has failed or is failing, you will soon have a dead 12V battery. It may be advisable to keep the new battery or a jump pack in the vehicle until the issue has been resolved.