Will Setting Interrupt Pin High Damage Bluetooth Module?
I'm not sure, if you can rule out the hardware damage (you wrote, that you removed the linear voltage regulator, but didn't explain more there). But with that code you won't get what you want.
I see multiple issues, all boiling down to the same problem: You cannot do that inside of an ISR:

delay() relies on interrupts to happen during its execution. As of my knowledge this leads to delay() never ending, blocking your code. Also it is very bad design to have a long running ISR. So much is depending on interrupts to work and when you have something long running, you can just easily do that inside the main code, triggered by an interrupt flag variable.
millis() will not increment during the run of the ISR.
delayMicroseconds() will work, but you still should keep the ISR very short. Reading an ultrasonic sensor is a no-go.
Serial.write() will fill the libraries internal TX buffer, but the actual transmission of the data also relies on interrupts. You won't get your data that way.

Instead of doing all that inside the ISR, you should just let the Arduino wake up, then set a (single byte) flag variable. In your main code you need to check for that flag with an if statement. In that if statement, you are doing all what you need to do and at the end you are resetting the flag and going to sleep again.