I am using a headset for calls but this headset is not on my head unless I am on a call. I noticed you have added the secondary ringer in the headset settings but I am saddened to see that this setting only applies to calls and not chat notifications.
I work on more than one computer screen and I don't always realize someone is chatting with me until I look at my taskbar and see numbers next to the Teams Icon, would it be at all possible to add the sound notifications on both headset and pc speakers as you did for incoming calls?
You may want to check the uservoice feedback forum to see if others have made this suggestion for the product. If not, feel free to make one yourself for others to vote on, and you can keep track of the progress. https://microsoftteams.uservoice.com/