An update:
I ran the MCPR removal utility, but I had a blue screen error while it was running. After rebooting in safe mode, I tried to run it again, but it wouldn't start.
Fortunately, after contacting their technical support, they were able to finish removing everything. Since then I've had 2 auto-updates for Avast and haven't had any issues. I'm cautiously hopeful that my problem is fixed.
Thank you to everyone who responded.