It would be helpful to know what codes were set when the transmission problem message appeared. I believe that you saw the "Transmission System Problem" indicator. When I had a similar issue, this flashing indicator took over the speedometer screen, and an alarm sounded. Definitely got my attention.That is just it. I had no issues with the car. I had a message that popped up that there was a transmission problem noted. The thing is that it went off right away and never came on again until a few weeks later. I tried to get it checked out but the first dealer, not a Honda dealer, thought it was a sensor issue and reset it or something like that. Later it came on again so I took it to Honda and I am not sure what code came on but they gave me the news that it was a big issue. It was not over heating so the extended warranty that was put in place for that issue did not help my issue. My car had just 63,000 miles on it but past the warranty. So Honda basically said I was stuck. I still feel Honda knew there were transmission issues with the 2016 and instead of any recall, the consumer is getting stuck with a huge bill.
The service garage where you first brought the car should have read the trouble codes set in the computer, and there would certainly be a code set by this issue. Their service ticket should show odometer miles, and the trouble code. If not, try to contact them about this. They did you no favors by resetting the code and saying it was not important.
In my case, this trouble alert happened to me at 58,800 miles. Code P0741 was set which indicates a slipping torque converter lockup clutch. The Honda dealer changed out the fluid and said it was fixed. A few days later, the alert repeated, and the dealer then replaced the torque converter under warranty. Interestingly, at first the dealer was going to replace the transmission, but then the Honda Tech Line called them back and said only the torque converter was required.
I don't know if your issue is the same, but I suspect it is. Perhaps you can document your initial visit to the non-Honda service shop and prove the presence of a P0741 code prior to 60k miles. Perhaps an appeal to Honda for consideration, maybe with some legal help, could result in them sharing costs.
The whole extended warranty thing confuses me. Powertrain coverage is the whole point of an extended warranty. Why didn't this cover your transmission?
You may need a lawyer to help you with this.