Well, this is embarrassing. When I did a quick subtraction of odometer readings after noticing there was even a difference, I got 12 miles, but after looking at my iPhone photos (I took pictures of the dash display since I didn't have a pen handy), I see that it's actually 312 miles, which is about how far I drove that day. So it must have been in "Ready to Drive" mode the entire time and didn't get enough airflow to keep cool. Fortunately, no damage was done.
I might just leave it off altogether (i.e. not in Acc mode) to help avoid this in the future. The manuals for the 2014 and 2015 C-Max both indicate this in the Recreational Towing section, so it's probably the same for the 2013, whose manual hasn't been revised in the last year.
By the way, I did install a charge line. It works great, with and without the engine running!