IF you are getting two-way communication as I have described, it proves that your code, components, procedure, and wiring are all kosher. And I'm sure that includes the voltage divider, which should be the least of your problems.
The message "Enter AT commands:" comes from Arduino and doesn't prove anything about Bluetooth.
I'm afraid I am at a loss to explain this, but it can only be procedural, and I can only offer two wild longshots.
!. You are trying to configure via bluetooth, i.e. the phone. You can only do it via serial.
- It's not an HC-06, it's an HC-05. I say this only in the light of recent thread on this forum, where it was pretty clear that was the case. The module was on a GW-040 breakout board, ahd four pins, and identified itself as "HC-06". A genuine HC-06 identifies as "LINVOR".