The 1851 census was taken on the night of March 30/31st, so John should have been younger than you thought when recorded there, to match his bapt on 2 March. I checked it on Findmypast, and he is actually recorded as "1m." on the scan, which fits his bapt date very well - usually not long after the birth at that time.
I don't think there is any doubt that this is your John Moore, although there was another baby John Moore in Patrick as well. (Just checked - he was aged 11 months, with parents Charles & Elizabeth Moore living at Kerowdhoo, Patrick.)

1851 census - household transcription:
Name Relation Condition Se x Age BirthYear Occupation Where Born
MOORE, William Head Married M 32 1819 Agricultural Lab Patrick Isle of Man
MOORE, Anne Wife Married F 32 1819 Patrick Isle of Man
MOORE, Anne Daughter F 9 1842 Patrick Isle of Man
MOORE, Margaret Daughter F 3 1848 Patrick Isle of Man
COLVIN, John Uncle Married M 50 1801 Agricultural Lab Patrick Isle of Man
COLVIN, Anne Aunt Married F 44 1807 Patrick Isle of Man
MOORE, John Son M 0 (1 Month) 1851 Patrick Isle of Man

Piece:2523 Folio: 10 Page:13
Civil Parish: Patrick Municipal Borough:
Address:
Creggan Mooar, Patrick, Isle of Man

It's a bit of luck that William Moore's aunt and uncle are staying with them, so that you can find the right parents for him.

In case you haven't checked this, he looks like WM MOORE Christening: 11 OCT 1818 Patrick Father: PHILIP MOORE, Mother: MARTHA CALVIN [IGI extracted entry]. An earlier William was bapt 21 SEP 1817 (presumably died?).
Also THOS MOORE Christening: 12 DEC 1824 Patrick (mother Martha COLVIN).

PHILIP MOORE & MARTHA CALVIN married 21 NOV 1816 Patrick.

Sue