Sounds like a good idea to me! This would give the astrologers a "spot" to put in the missing constellation on the ecliptic (Ophiuchus).
We would then have 13 "Moonths" in a year. It would also mean, if it was New Moon on the first of the month, it would be new moon on the next month...it would just slowly "walk" through the calendar as a month and moonth don't exactly coincide. If New Moon was on a weekend, it would be around the weekend for several months...allowing the best viewing to be on the weekend for a year or more at a time.
They could just change the old rhyme to:
28 days hath September, April, June and November. All the rest have 28, except February (which is also the last month of the "new" year) which has 28 but 29 every leap year, but it is a holiday anyway so who cares!
I just don't know how the triscadecaphobians will handle it!