Interestingly, the -fere in interfere comes from the Latin ferīre, meaning 'to hit', 'to strike', etc. My first guess would have been something like facere/fāre or -fer, but that quickly falls apart on reflection (to do across? between-bearer?).

Inter + ferire = to strike one another. Makes sense.

Bonus point: the aforementioned -fer ('bearer', like conifer or aquifer) is distantly related to ferīre, as it is to English to bear, Greek phérō ('to carry'), Slavic brat ('to take'), Sanskrit bhárati ('to carry'), etc. I suppose ferīre itself must be the result of semantic drift along the lines of 'to carry/bear' -> 'to bring forth [blows]' -> 'to strike/hit'.