In addition to the familiar September–December, there were two more numerically named months before they were renamed in early imperial era: Quintilis and Sextilis.
These should definitely go to your slots 5 and 6.
In English you could call these Quintile and Sextile.
You seem to have slightly misanalyzed the ending.
What you add to the end of a number is not -ember but -ber.
This is an easy mistake to make, as three of the four usual month numbers (septem, octo, novem, decem) end in -em.
The key observation is that we have October, not Octember.
It appears that the numbers 7–10 get the ending -ber to the cardinal number, whereas 5–6 get the ending -ilis to the (stem of the) ordinal number.
That is, Quin(c)tilis and Sextilis seem to be based on quintus and sextus, not quinque and sex.
So, there are two attested ways to produce month names from numbers.
To me it sounds more natural to use the ordinal one for the first months:
- primus > Primilis
- secundus/alter > Secundilis/Altrilis
- tertius > Tertilis (somehow Tertiilis sounds unnatural)
- quartus > Quartilis
Tertilis and Quartilis (or Tertile and Quartile in English) sound quite natural to me, but the first two less so.
However, they feel more understandable and Latin to me than the ones you proposed.
With this pattern the first half of the year has -ilis, the second half has -ber.
I find no way around the somewhat clumsy Undecember and Duodecember.
As Draconis points out in another answer, the Latin numbers 11 and 12 are undecim and duodecim, so maybe you could change the month names to Undecimber and Duodecimber.
However, it would not be unreasonable to keep the -e- by analogy in the English versions.
I don't know whether adding -ber in Latin would change the vowel or whether later developments in English would naturally lead to a vowel change, but this is just a minor detail.
One practical benefit of the -i- is that it sets the eleventh and twelfth month apart from December.
The names undecimber and duodecimber are used for the thirteenth and fourteenth month in some contexts, such as computing.
An option worth serious consideration is having the year start in March instead of January.
This is what the Roman calendar did, and this is why the names are off by two.
This would have the benefit of the months names aligning with the ones already in use.
Here is the suggested list in English resulting from the considerations above:
- Altrile or Secundile
- Undecember or Undecimber
- Duodecember or Duodecimber