Colin
Richard Ballard gives his birthplace as Westmean Hampshire on the 1861 census circa 1806. He died just after the census. The nearest to Westmean would be West Meon - not far from Ropley but a long way from Ringwood. In 1851 his birth age is consistent circa 1806. I don't know his age on death as it is to early in the death register for ages.
There is a Peter Ballard buried in the West Meon registers in 1800 - 1st April.
Hampshire has poor coverage on the IGI only about 43% parish coverage pre 1813. The BVRI supplements it (often not from the registers themselves but from the Bishops Transcripts which leave gaps - but with Ropley I think the baptism register is missing for the crucial period - Hampshire Record Office would be able to confirm that)- no coverage of West Meon on either. Hampshire Genealogical Society have a baptism index on CD but only from 1813 onwards.
There are a lot of Ballards in Ringwood on the censuses and as yet there really is no evidence which proves they have anything to do with the Ballards you are interested in. After the baptisms you have mentioned the series continues in Ringwood with John's wife named as Sarah.
e.g.
JOHN BALLARD
Christening: 20 DEC 1812 Ringwood, Hampshire
Father: JOHN BALLARD
Mother: SARAH
You also have no evidence that Louisa was baptised in Kilmeston. Goddard is not a rare surname and you would expect at least some John and Mary Goddards to be having children in Hampshire at this time.
At this point you have only three potential earlier born children - pre Sophia Jane and Fanny in Winterslow.
Richard born circa 1806 - possibly West Meon
Ann born circa ? (no evidence as yet to believe her name was anything other than Ann) - still to eliminate or confirm the Titchfield possibility for a birth.
Louisa - you don't specify her age born circa ? (have you found her on the 1841 census - does she indicate not born Wiltshire?)
There is no indication as yet that where ever and when they were baptised they were not baptised in the name Ballard.
Did Richard marry as Ballard?
Louisa married as a Goddard (is her father's name on the marriage entry given as Goddard or Ballard?)
Marriages Mar 1842
GODDARD Louisa Alderbury 8 349
HATCHETT William Alderbury 8 349
We presume but as yet have no evidence for that Ann married as a Goddard circa 1832.
Was John buried as a Ballard? His death seems to have been registered as such.
Deaths Sep 1843
BALLARD John Alderbury 8 152
If he was buried in 1843 where was he on the 1841 census - and Richard and Louisa?
Does John's burial entry give any further information?
Therefore at this point the known period of Goddardness starts post Fanny's baptism in 1821? By 1830 at least if Richard married as a Goddard not a Ballard? And lasts until certainly John's death in 1843 when it becomes more dual and fades completely by 1851 when it returns to Ballard. (Richard's family were Ballards in the 1841 census so they are the exception - are their children baptised as Ballard or Goddard?)
The most likely reason for the Goddard alias would be that John was born illegitimately. At this point there is no reason to believe John did not marry Mary and their children were not born legitimately.
Have you tired the Hampshire Genealogical Society's marriage index for potential John Ballard marriages to Mary in the Ropley/West Meon area or perhaps John Goddard to Mary?
I think the index is probably only available to search for members of the Society (which is by the way an extremely good one - with a very informative quarterly journal).
http://www.hgs-online.org.uk/Regards
Valda