Komban wa, Flaminius San to Shiremono San.I looked it up in some ordinary Japanese dictionaries. They simply introduce two opinions about the derivation; [目門 ma(eye)+to(door/gateway)], [間戸 ma(space)+to(door/gateway)].
So this etymology has already been contemplated by Japanese scholars, and the only problem is about the meaning of each component.
By the way, I have found another Yamato compound in which -e> -a: kaze "wind" + kuruma "wheel" > kazaguruma "windmill", so me > ma- "eye" is not impossible.
I suppose there are many more compounds of this type.
This thread has been branched from an etymology discussion that supposedly involves vowel gradation. The topic of the current thread is Japanese vowel gradation.