Did both parents emigate as neither appears in the 1871 census.

1861, Douglas, district 3, schedule 13, 3 Great George Street
Robert Moore, head, married, 28, tanner, German
Jane, wife, married, 24, dressmaker, Braddan

1841, German, district 2, page 2, Knocksharry
Thomas Moore, 55, agricultural labourer
Eleanor, 45
Thomas, 34, agricultural labourer
William, 10
Robert, 8
Margaret, 5
Robert, 1

I've double checked and the two Roberts are as shown.
Robert (8) bp 30 Dec 1832 is your man.
Robert (1) no bp found but appears aged 11 in 1851 as son to Eleanor, widow.

1871 census has:
Margaret Jane with grandparents Daniel & Elizabeth Cowley
Catherine Ann with cousins William & Catherine Quiggin
Robert Stephen with aunt/uncle Richard & Elizabeth Cavendish

Children of Thomas Moore & Eleanor Kneen include:
Jane, bp 3 Aug 1823
Elizabeth bp 12 Mar 1826 (?m Cavendish)
William bp 4 Jul 1830
Robert bp 30 Dec 1832
Margaret bp 6 Dec 1835