Are you aware that Harriett had another illegitimate child before George.
This is her birth registration
GAME, ELLEN - (no mother's maiden name indexed)
GRO Reference: 1841 D Quarter in RINGWOOD Volume 08 Page 195
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I have only looked at this quickly, but it does look as if James James did have a son named
John James, from his first marriage to Elizabeth.
John was born in 1813 in Christchurch. He was married in 1834 Ringwood to Martha Wright.
In the
1841 census he is living in Kingston, Ringwood with his wife and 5 yr old son Thomas. He is an agricultural labourer.
https://www.familysearch.org/ark:/61903/1:1:M79C-9Q6John's abode is very close (on previous census page) to his father James who is with Mary and Harriett Game and 1 yr old son named George. (This son, George died age 3 in 1842)
https://www.familysearch.org/ark:/61903/1:1:M79C-97TIn the 1851 census John James is living in Ringwood with his wife and children. He is now a cattle dealer.
https://www.familysearch.org/ark:/61903/1:1:SGPK-CYYSo there is the potential that John James was indeed George’s father, and that George knew who his father was. I see that George himself was a groom / ostler/ horse handler, which is how he describes his father’s occupation.
If his father’s name is not on George’s birth certificate, then DNA matching would be the only way to be certain.