If this is the couple you are looking for, then they also had the following:
Joseph bpt 16 Jun 1788
Elizabeth bap 12 Oct 1789
William bpt 1 Jun 1801
Thomas bpt 27 Dec 1805
all baptised at St Mary's.
I think these are also theirs:
James bp 29 Jul 1792
Sarah bp 18 Jun 1794
Mary bp 30 Sep 1798
recorded as BARNS, at Millbrook.
I think this may be the marriage:
Joseph Barnes married Elizabeth Lambert 5 Aug 1787 at Holy Rood, Southampton
Sorry - couldn't oblige with Mowatt!

Nell