Or, we use Latin names of numbers. So, Unusber, Duober, Tresber, Quattorber, Quinber, Sexber [rename as necessary for America], September, October, November, December, Undecimber, Duodecimber, Tredecimber.
These would be better names from a purely phonological point of view: Unusper, Tresper, Quattober, Sexper. That way you don't have awkward clusters of both voiced and unvoiced consonants. The -ber affix having originally come from mensris (-mens-ris > *-membris > -ber), devoicing the initial 'b' to 'p' in those circumstances is reasonable enough. 'Quattober' is a simplification that makes it fit the pattern better.
Alternatively, you would rely on the ordinals, dropping the -us ending and replacing it with -ilis as in Sextilis and Quintilis.
Alternatively, you would rely on the ordinals, dropping the -us ending and replacing it with -ilis as in Sextilis and Quintilis.