Hello and welcome to Apple Support Communities ashackel,
Thank you for contacting us. You have come to the right place and we are happy to assist you. We understand you would like the alarm not to play after you fall asleep. We can understand why you’d reach out for a solution and actually that was the topic of a conversation recently. To start, make sure you have the most current Operating System on your device (Settings > General > Software Update). There are a couple work arounds that you can try but no way to just change the setting to "not" play via bluetooth.
Adjust audio and visual settings on iPhone - Set headphone accommodations
Play audio using Siri on HomePod - Set a sleep timer: Stop music, podcasts . . .
Add an alarm on Apple Watch - "You aren’t alerted on your iPhone when your Apple Watch alarms go off."
Good luck, hope this options help!
Thank you for contacting Apple Support Communities.
Best regards.