The service terminating unexpectedly is nothing to do with avast, it scans and alerts to infection, it doesn't block or take autonomous action.
Is this the free version or the paid version of MBAM ?
Presumably it is the paid for option as I have the free version and there are no MBAM services running, when launched mbam.exe is running, but no services.
There are some avast forum members also using the paid option with no problems, as I said this is the only post I have see where someone with MBAM has experienced any problem.
You can stop avast on-access protection but it is designed as a resident scanner and on reboot the program would start again.
All I can suggest is uninstall MBAM, boot into safe mode and install it from safe mode (avast doesn't run in safe mode) and then try to start the MBAM protection in safe mode and see if the same problem happens. It is quite possible it will work, but that doesn't pin down what the true cause is as lots of things don't start in safe mode, that is the whole idea of safe mode.
You could also try uninstalling avast and reboot and see if the MBAM problem still exists, if so uninstall and reinstall MBAM.