Just to update this thread, I have another Obi200 connected to GV with another Panasonic cordless phone (different model from the phone mentioned in the original post), and the communicating base for this phone is connected directly to the Obi rather than to the house wiring. If I enable the "Use This Service for a Security Alarm Line" option in the settings for SP1 (the GV account), just as I had for the other Obi device that wouldn't respond to button presses in GV Voicemail, it automatically changes the DTMF Method to InBand in the Obi Expert configuration. Upon doing this, GV Voicemail prompts fail to respond to button presses, same as the other Obi setup.
If I disable the "Use This Service for a Security Alarm Line" option, the DTMF method is automatically set to Auto and GV Voicemail correctly responds to button presses with this setting. Which method does the "Auto" setting actually utilize? Does it intelligently pick one based on the service provider being used? Does it always just use RFC2833?
I guess it comes down to the Panasonic phones failing to properly communicate with GV Voicemail when using an InBand DTMF Method, even though it seems to work with other prompt-driven systems (though I have discovered one other system that fails to respond to button presses over the last week). Or maybe it's due to Obi's (or GV's?) handling InBand tones?
EDIT: Did a little searching and found this thread. Looks like this isn't a new problem:
https://www.obitalk.com/forum/index.php?topic=6281.0