Hello,
First, thank you for building our project!
This is very unusual behavior. There is typically almost no lag at all in the controls, and the intermittent cut outs are also not typical.
Let's talk about the 9v battery first. The 9v output should go to GND and Vin on the nano. The HC05 and SD card reader should be taking power from +5V outputs on the nano. In other words, they are using the nano's internal 5v regulator, which can just provide enough power for those two modules. So, carefully check the wiring diagrams and make sure you're not feeding +9v directly to the hc05 and SD reader.
Okay, now the laggy/stopping issue. I can think of three reasons this might happen:
1) Some 18650 batteries have an aggressive protection circuit that will cut power if there is a surge in amps. When motors start up, there is such a surge. You might have batteries whose protection circuits are cutting out power intermittantly. However, arguing against this possibility is that you're not losing bluetooth connection (it would take longer than a second to reconnect.)
2) A loose wire to the servo driver could cause it t loose power. It would then reboot, and start working again. Check power connections and also the signal connections (SCL/SDA) to the servo controller. (Arguing agains this is that it works on demo mode, but it's still something to check).
3) The BEC you are using for power conversion might not be able to handle the amps. It might have protection circuitry that cuts it off automatically. It would then reboot and start working again.
None of these theories really explain the laggy behavior, though, I'm scratching my head on that one. But let's see if fixing the drop-out might also fix the lag.
Some questions for you:
1) Send me a picture of your BEC, including any markings or wording or model numbers showing on it.
2) Send me a picture of your battery, showing any labels/markings/model numbers.
3) Did you install a buzzer? If not, it might be helpful to hear the output from the buzzer as there are several situations that cause different kinds of beep that are useful for debugging. I do not hear any beeps which could be either because none of the common error conditions are occurring, or because you didn't install a buzzer.
4) The video shows you using "scamper mode". Does the same thing happen on slower walking modes like W1? Scamper mode draws far higher amps than normal walking, so if it only happens in scamper mode that would point to the batteries or the BEC.
Also, a picture of your hc05 would help, perhaps this is a radio issue that I haven't seen before.
Hope this helps,
Steve P.