The response that came to mind when the question first occurred to me was "Perhaps the intended meaning of the clause is 'He is the image of the otherwise-invisible God'". This is similar to how one might say "the healthy boy with the sprained ankle" to refer to a single boy who is very healthy overall.
However, this explanation is ad-hoc, and it certainly isn't the literal meaning of the verse. Thus, in my opinion, this explanation bears the burden of proof--proof that the words or grammar are poorly translated, or that the verse isn't intended to be strictly literal, or something else.
I believe that I found the proof in a very observant answer to a tangential question about the same verse. Its author, Dottard, notes that the phrase "image of the invisible" is an oxymoron; invisible things don't have images. To me, this is sufficient evidence that Paul is using language loosely to grant credibility to the explanation above.