Hello Hoosiers16,
We have a couple suggestions on how to approach the problem of Bluetooth disconnecting. Be sure to test if the issue is resolved after each step.
First, update to the latest iOS version. Just as a reminder, here are the different ways available to update iOS: Update your iPhone or iPad - Apple Support. Be sure to follow the instructions on how to make a backup of your data before updating.
Next tap on Settings > Bluetooth and toggle Bluetooth off, and then back on.
Power cycle the iPhone using these instructions: Restart your iPhone - Apple Support While simple, this can refresh the iOS, apps and connections.
Restart the Bluetooth device that is disconnecting.
Test device that disconnects with another Bluetooth device to isolate the issue correctly. We want to see if the device disconnects from other units or if it stays connected.
Confirm that there is not any interference: Resolve Wi-Fi and Bluetooth issues caused by wireless interference - Apple Support
Disable Wi-Fi by tapping Settings > Wi-Fi and toggling it off.
If you are still having an issue, Apple Support would be out next step: Get Support
Cheers.