I don't think it's Bluetooth. For me it was related to iMessage.
The Messages app combines iMessage communications and SMS messages. Apple created iMessage when mobile carriers charged for SMS messages. iMessage will work across multiple Apple devices. I have had cases where someone will send me an iMessage and my iPad will accept the message. Then I never see it on the phone. If you only have an iPhone and no other Apple device, iMessage is useless. It's a PITA. You don't need it.
FYI, in Messages, SMS messages will appear in green and iMessages will appear in blue.
First, sign out of iMessage. Go to Settings -> Messages -> Send and Recieve
Click on the blue Apple Id line near the top
Select "Sign Out"
In the Messages settings (Settings->Messages), turn off iMessage.
After turning off iMessage, if you have friends with iPhones, you may get an error when sending a message to their iPhone. Reply to them in Messages using their mobile phone number.
All my Bluetooth devices work fine. Turning off iMessage won't hurt anything and it may fix your problem.