This is a great question! Setting aside other uses of "a bit" ("a bit of luck", "it hurts a bit"), it seems that as a general rule, it can only modify an adjective or adverb that is functioning as a predicative complement.
- Your accent is a bit/a little/somewhat strong.
- Your accent sounds a bit/a little/somewhat strong.
- The actor made his character's accent a bit/a little/somewhat strong.
In all of these examples, the phrase comprised of modifier + strong is the predicative complement of the clause.
The adjective may be a comparative adjective, in which case "a lot" also works:
- Your accent is a bit/a little/a lot/somewhat stronger now.
- Your accent sounds a bit/a little/a lot/somewhat stronger now.
- The actor made his character's accent a bit/a little/a lot/somewhat stronger than his own.
Except for "somewhat", these degree modifiers do not work in other parts of the clause:
- A somewhat strong/stronger accent is necessary for the part.
- *A bit/little strong accent is necessary for the part.
- *A bit/lot/little stronger accent is necessary for the part.
Or, to craft an example more like the one of the original question:
- The trick is a somewhat strong accent.
- *The trick is a bit strong accent.
While a somewhat strong accent is a predicative complement, somewhat strong by itself is not! It is an attributive modifier of accent.
I suspect that the historical explanation for this restriction has to do with "a" being the indefinite article. (Replacing "a bit" directly for "somewhat" gives "a a bit strong accent". Or maybe it would become "an a bit strong accent". Either way, this makes no sense—the indefinite article doesn't repeat in English!)
From a corpus search I found some examples where the modified word might be considered an adverb (or something other than an adjective). Paraphrasing slightly:
- They begin to take their personal metaphors a bit too literally.
- The new provision reads a bit differently from the old one.
- The technique makes the scent of perfume last a bit longer after each application.
- Miramar, California is a bit north of San Diego.
There is another possibility, which is modifying quantity terms like "more" and "too many"—here there is no restriction to predicative complements:
- I ate a bit too many cookies.
- A bit more food is necessary for the party.