While you're right that transversus is masculine and transversarium is neuter, the latter is not genitive, but these are actually two different words.
The first can be broken into three parts: trans-, vers-, and -us.
The second, however, into four: trans-, vers-, -ari-, -um.
The first is a pure past participle from transverto ("to turn away"). Here you would expect that the processus itself turns away.
Transversarium though adds the adjectival infix -ari-, which denotes a relationship to the verb transverto. This relationship could take a few forms, but the basic one is that it allows something else to be transversus.
Now, medical terminology is highly idiomatic, and I'm not a surgeon or medical historian, but if this website is accurate and I'm understanding it correctly, I think this bolded part explains the meaning:
The transverse processes are each pierced by the foramen transversarium, which, in the upper six vertebræ, gives passage to the vertebral artery and vein and a plexus of sympathetic nerves.
It thus allows the vertebral artery and vein to "traverse" what I'm assuming are the transverse processes, though again, I am not a doctor, so hopefully someone with more medical knowledge can explain the action in greater detail.
A quick note on endings, the neuter genitive plural of transversarium is actually transversariorum (a mouthful!). transversus and transversarium are both 1-2 adjectives, and therefore follow the same inflection pattern, which you can see here. Note that the second -s- is a part of the stem, not a part of the ending.
