Question before proceeding

I have read the guide and I have one or two questions as I am not sure I understand correctly. I don’t want to waste a whole lot of time looking for documents if it’s a no go from the start.

I am looking into citizen by descent from my Great Grandfather.

Born: Willmersbach Bavaria June 29, 1896. Mother born 1872 - Bavaria? All censuses just says Germany Father born 1866 - Bavaria? All censuses just says Germany

My GGpa, his mother, and his 5 other siblings immigrated in 1906, then naturalized in 1910. Per those same censuses. However, his father states he immigrated in 1883, and naturalized in 1887. Went back to Germany at some point, married in 1895/6. All children born in Willmersbach (last kid was born 1903) and all went to America in 1906. I found a ship manifest with that date. And the only one I could find. Other than possibly GGGPa in 1883 from a Hamburg list (I think?).

So, is this a possibility that I would be able to have citizenship by descent since GGpa was a minor & his mother was German or is it not possible because his father was naturalized (possibly?) in 1887? I suppose I could at the very least see if naturalization documents are available to confirm timeframe/validity. A census could be wrong for whatever reasons.

Thoughts? I’ll try and answer any clarifying questions the best I can.

TIA