Can you provide more specifics on what you have control over? Are you a full stack developer who is writing the code for "dbNotifierBETA"? Or are you just someone who is trying to integrate with it?
If the Teams App "dbNotifierBETA" is a bot, then the developer (you?) should be able to implement something like what is discussed here:
"Channel and group chat conversations with a bot"
Andrew Connell is a great resource for all things MS Teams app/bot. He is the person shown in the video on the above page.
Then your Power App should be able to use the Teams action to send a message to a channel that the bot could pick up.
If you were to go the route of creating an HTTP endpoint in your bot to send messages to, then you have other means of completing your goal besides Power Automate. You could use a Logic App which runs on a Consumption Plan instead of a Power Automate flow. An HTTP Action in a Logic App works differently that Power Automate in how MS charges you for the feature. On a Consumption Plan, you are charged (I am simplifying here) by the number of times your flow runs.
Any specifics you can provide would help.
1) Do you have the ability to make changes to the "dbNotifierBETA" app? If not, can the developer provide you with any information such as does this bot listen to a channel for @mentions?
2) Where are the messages coming from that you want to relay to the bot? An inbox? A Table in SQL? A SharePoint List? Can the bot/app be created to check that location directly instead of the need for a Power Automate flow to do that task?
3) Currently what are you able to do with this bot? Are you able to converse with the bot as a user? Are you able to say for example "@dbNotifierBETA here is the body of an email I received....."?
You mentioned the concern for using a Premium Connector but depending on how you accomplish what you are trying to do, you may only need a license for a Service Account that the flow runs under.
Any additional information you can provide would help.
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Thanks
-Mark