I don’t know of any books I can recommend, but I’d definitely be down for 13 months with one being short. We could do 12 months of 30 days each plus a 13th month of 5 (or on leap years 6) days.
As far as anything that exists today, there is the Unix Timestamp which is defined as the number of seconds since (the entirely arbitrary time of) midnight January 1st 1970 UTC. Of course, “1970” only makes sense in the context of the Gregorian calendar which still has to do with the birth of Jesus. So, it’s not exactly what you’re looking for. But maybe it’s at least more removed from “the birth of Jesus” than the Gregorian calendar we all generally use.
I guess if you’re interested in this stuff, you might be interested in learning about ISO-8601, a standard way of representing dates/times in text. And also the concept of “leap seconds” and things like Leap Smearing.
There’s also a great short story about someone trying to explain to an alien with no familarity with earth how our calendar works, but I’m having trouble finding it now. I’ll edit this post with a link if I can find it.