there's a possible baptism for Elizabeth, oldest daughter of William & Elizabeth. Her birth was registered sept qtr 1868 Portsea Island and the Catholic baptism in St John's Cathedral, Portsmouth is for Elizabeth born 26 July 1868, bp 5 Aug, daughter of William Clayton & Elizabeth Hardwick, no maiden name given. One of the sponsors is a James Hardwick.
There's a parish marriage at Portsmouth 26 Jan 1863 for James Clayton Hardwick and Naomi Swailes, both of 6 Tower Street, both of full age, he is a shoemaker, father James Clayton Hardwick, also a shoemaker, her father John, a tinman.
1861 census Winchester Street, Salisbury RG9/1316 folio 113 pg 26
James Hardwick head mar 49 cordwainer b. Kingston upon Hull
Elizabeth wife 43 b. Sutton Ashfield
James son 20 cordwainer
Mary dau 15
William son 10
Robert son 8
Charles son 4
all children born Hull
However the births of the children are registered in various places
James Clayton Hardwick sept qtr 1840 Hull vol 22 pg 391 - bp 9 Aug 1840 Sculcoates RC
Anthony Clayton Hardwick march qtr 1843 Hull vol 22 pg 422 - death reg june qtr 1844 Hull
Mary Ann Hardwick june qtr 1845 Hull vol 22 pg 252
William Clayton Hardwick march qtr 1847 Northwich vol 19 pg 193
Robert Clayton Hardwick june qtr 1851 Reading vol 6 pg 252 - he was born 14 June 1851 and baptised 16 November St Michael & the Holy Angels RC West Bromwich
Charles Clayton Hardwick sept qtr 1856 Worcester vol 6c pg 258
Frederick Clayton Hardwick march qtr 1854 Belper vol 7b pg 423 - death reg dec qtr 1854 Potterspury
mother's maiden name on all the registrations is Monsier/Monsieur
James Clayton Hardwick married Elizabeth Monsier 26 April 1836 in St Peter Ad Vincula, Stoke on Trent
The son of William & Isabella was registered as James Clayton Hardwick march qtr 1877Poole vol 5a pg 294 mmn Reay
With the way this family moved around, it's no wonder William was unsure of his place of birth.
One problem with this is that there is a death reg for William Clayton Hardwicke aged 48 dec qtr 1894 Poole vol 5a pg 156
So, perhaps the James that Isabella is with in 1901 in Bath is William's brother. His age is nearer that of James. Death reg James Hardwick aged 64 dec qtr 1906 bath vol 5c pg 313.
Previous census for him as a hawker in Poole 1881 RG11/2095 folio 111 pg 33 with wife Mary. No sign of Naomi after the marriage, no remarriage to Mary that I've found so far...
I suspect that Isabella didn't marry William or James, there is a possible death reg in Bath
Isabella Ray age 79 march qtr 1931 Bath vol 5c pg 690