If your TQFT does not extend to the circle, then the answer is no. A reference is: https://arxiv.org/abs/1408.0668

The proof goes by an explicit generators and relations presentation of the 3d cobordism category in terms of surgeries and automorphisms of surfaces.

If your TQFT does not extend to the circle, (EDIT: and you're willing to generalise to lax TQFTs,) then the answer is no. A reference is: https://arxiv.org/abs/1408.0668

The proof goes by an explicit generators and relations presentation of the 3d cobordism category in terms of surgeries and automorphisms of surfaces.