Is it an overkill to add external integrity check to the AES-GCM encoding?
Theoretically, there is no issue adding some kind of MAC on top of authenticated encryption's builtin. However, in practice there might be subtle flaws with composing the particular primitives you're using, or you may make an implementation flaw that renders them both vulnerable to a side-channel attack that didn't exist previously. Ultimately, it's best to stay away from getting creative with protocol design - just use the best-known implementation of the best-known primitive for the job.