I'm afraid your construction is not correct: it should be something like a bit of a different reason. You can see that a bit different [noun] is not used in print, whereas a bit of a different is.
There is a different reason, and we take a bit of it. It is a noun phrase (a bit) modified by another noun phrase (a different reason). That is a bit different is another construction: there a bit modifies the adjective different directly, and there is no noun phrase. You can't easily combine the two, so the slightly odd a bit of a ensues. A bit of a different reason may sound slightly like a bit different reason in quick or sloppy speech.
The construction is odd, but it is what I would call an explicative or defining genitive: you characterise the different reason as "a bit", as a small thing. It is akin to he is a bear of a man; that is not about a man possessing a bear, but about a bear that is a man.
See also this question on Linguistics: possessive constructions signifying semantic equation: term? distribution? explanation?