I have sat down trying to make some further sense of this Lelu. I think you are right and the William camped with Uriah Cooper, his uncle, on Mitcham Common in 1881, is the son of Aaron Smith and Cordelia Cooper, and therefore grandson of Edward Cooper who is also camped there that year. That seems to point towards Edward Cooper, born circa 1811, as an elder brother of Uriah and Spicer, and there is a gap in the baptisms of William Cooper's children where he would fit well enough:
Riley Cooper, son of William Cooper ‘A Gipsey’ and Chesiah, baptised at Clewer, Berkshire on 21st December 1800.
Connsaletta Cooper, daughter of William Cooper and Lucy, baptised at Brasted, Kent on 27th March 1803.
Hannah Cooper, daughter of William Cooper and Hannah, ‘Travellers’ baptised at Tudeley, Kent on 11th August 1805.
Joannah Cooper, daughter of William Cooper and Hannah ‘Gypsies’ baptised at Little Bedwyn, Wiltshire on 25th December 1809
Spicer Cooper, son of William Cooper and Anna, baptised at Greenwich, Kent on 2nd April 1815.
Uriah Cooper, son of William Cooper ‘Travelling Brazier of Woodside’ and Lucy, baptised at Croydon, Surrey on 11th July 1817.
What also points in this direction is the lasting link with the Junix family. Uriah Cooper is camped with his wife and child next to Joseph and Kezia Junex in tents at Holty Common, Hartfield, Sussex on the 1851 census. I think it is fairly likely then Kezia Cooper (b.1796 circa) partner of Joseph Junnix, is an elder daughter to William Cooper by his first wife Chesia/Kezia. The fact that Levi Cooper is so strongly linked to all these families in the records might point towards his father James Cooper (b.1798 circa) partner of Harriet Lee, as another son of William. If James Cooper also had multiple partners he might be the father too of Margaret Cooper, partner of Moses Mills.
A strong link between James Cooper and Samuel 'Norwood Sam' Cooper (b, 1794 circa) partner of Floriana is speculated on in the J.G.L.S etc, but the fact his son Robert baptises a child at Ham the same day as Edward Cooper baptised his adult children, might be the extra evidence needed to indicate that he was also an elder son of William Cooper, probably by first partner Kesia.
The above might be speculative but all fits together well enough. The only problem I have is where Matty Cooper comes in. He also seems very strongly linked to this group for over a century, being camped with them , his children marrying with them etc. But he is usually down as a son of Harry Cooper (born circa 1780) and Mary 'Asha' Buckland.