If you provide 5v USB power to the RPi, it should be a stable supply rated for up to 2A (or more). Other than running the motors, the BrickPi3 can be powered and fully functional running off the 5v supplied through the RPi.
If you provide 7.5-12v (ideally 8-10v) to the BrickPi3, you can run motors, and the BP3 will provide a regulated 5v supply for the RPi.
It is safe to use both a USB power supply connected to the RPi and a battery pack connected to the BrickPi3. One advantage is that the RPi won’t brown-out or crash when the batteries get too low. Another advantage is that it could provide some of the power, taking some load off the batteries, and thus extending the battery life. For a stationary robot (or even a mobile robot sitting at your computer while programming), with USB power connected, you can safely switch off the BP3 power switch to electrically disconnect the batteries and save battery life.
For a mobile robot, I would recommend just using a battery pack connected to the BP3, and let that provide the 5v power for the RPi. It’s simpler than setting up a secondary battery bank to supplement with USB power. You can use rechargeable batteries or a rechargeable battery pack to power the BP3, such as this rechargeable battery pack.