Trish, Jenn and Genni,
Wow! I'm beginning to get just a little excited by the prospect that this individual ticks so many of the boxes. I realise that William Carter is a name that occurs frequently, but there are many details that fit. Certainly a Lydia Tidmarsh CARTER marrying in Tasmania in the 1840's simply has to be his daughter, thus connecting him with the island. He was most definitely in Melbourn (sic) Victoria in 1869 according to his brother's will, and this man you say died there.
My William CARTER was born on 7th May 1797 and baptised 4th June 1797 at St Botolph without Aldgate, London, so that is not exactly right. (But not that far out) Father was indeed Robert, but the fly in the ointment is the informant of his death not putting down that Lydia Tidmarsh was his mother.
You've all been so industrious while I've slept this other side of the world, and if you could possibly look for his will or any newspaper reports of his death, Genni, I would be more than grateful.
Meanwhile I'll contact the Tasmanian archives/library and see whether they have more on this individual.
If he turns out to be the right person and he indeed went out to Tasmania as a criminal, got his freedom after serving his sentence, then established himself as a merchant/trader like his successful father and elder brother, that would be quite some story...
Very best wishes,
keith