CPU minutes are a confusing topic, but generally one of the only things that's limited in shared hosting.
While your site might not be using much via traffic, depending on how often the cron runs that itself could be using up alot of CPU resources.
I would monitor it for a few days using the info at the end of the article linked above. Then you can try adjusting the frenquecy of the cron perhaps? If the cron is really the thing that is eating alot of CPU resources, changing the cron from 15 to 30 may halve the load and bring you back into tolerance.
When I read your first message yesterday I wondered if cron was still running on the server (it's possible but usually unlikely that the webserver and the panel can get out of synch), support should have checked that tho I would think.
The think about the CPU time issue that support pointed you to, is there should be a period of time each day that it works, then at some point you should go over your limit and start getting the weird errors. I would think there would be some consistency in the time of day too.
Once support sends you the first reply to a ticket, be sure to REPLY to that reply rather than opening a new ticket. Supports time to first reply is high, but after that successive reply time on the same ticket are often much shorter.