It would seem that the source of the marriage information is an abbreviated version through LDS ie. so I am assuming that you have not seen the certificate. That is why I was asking for the source of this information, not just a repeat of it.
You need to get this marriage certificate.
Tasmanian marriage certificates from 1896 have very useful information -
place of residence, usual residence, year in which the person is widowed, children (number living, number dead), father's occupation.
With the year of his wife's death, and if it is recent to this marriage, you know to look for a Ballard death in that year, and starting in the place of his usual residence.
The witnesses, and their addresses, might be Ballard relatives or his young adult son/daughter, or in-laws - relatives of the first wife.
If father and son have the same occupation, then other male Ballards of similar age (to James), similar occupation, might be related (death notices NSW or TAS for "...brother of James"). Is it a Ballard family occupation?
Is his mother Sarah Ballard, or just Sarah ie he does not know her maiden name?. If an adult does not know mother's maiden name, it might mean that she died at a young age and James has limited child/young adult memory of her - a clue when looking for his origins in NSW.
I would keep an open mind about his age - he is quite a bit older than Kate. Human vanity being what it is, he might have taken some years off to arrive at his stated 39 yrs.
You have very little to work with here to locate your James Ballard so you need to throw a wide net - and the first marriage would be good to locate. Good luck.