Personally, I would go for naming them after the twelve largest moons in the solar system, since they were originally (and still are in many cultures) based on the lunar cycles, giving the months Ganymede, Titan, Callisto, etc.
What would you rename the months?
edit: Simply using the numbers is already a thing (see, for example, 2026-01-01. January is represented as month 1), and reordering the existing month names would be a real headache, as it would be difficult to tell whether you are reading a date using the old or new naming scheme.
If you do that in Latin, you get names like September, October, and December. Just make sure not to randomly insert two extra months, making sevenuary by the 9th month.
The idea that July and August were added and shifted the months to be wrong is actually a misconception afaik. The months they replaced were already basically “fiveuary” and “sixuary”.
The numbering got shifted off when they decided January should be the first month; it used to be March. February was the last month, which explains why leap days are added to February.
Monuary, Dotuary, Trituary, Quadtember, Quintober, Sextember, September, October, November, December, Undecember, Dodecember