I see a few mentions with that name on TROVE (Have we looked for this before?)
There seem to be three generations of men with the same name however mentions from the 1800s would appear to be the one you are referring to:
1848 - referred to as "Dr Barker Benson" at a meeting near Gosford
1867 - a few mentions in the Insolvency court
1902 - his son's death notice has "Barker Joseph BENSON MD"
1871 - his funeral notice "Barker Joseph BENSON Esq, MD"
1842 - a person mentioning an advertisement supposedly inserted by him - I could not find the ad referred to.
The most interesting, though, is an account in 1849 of 'a Mr, otherwise known as Dr Barker Joseph Benson, a recent importation from the English Sawbones Nursery" to be examined with regard to a debated promissory note. He was committed for trial on embezzlement. This is from Bell’s Life in Sydney and Sporting Reviewer (NSW : 1845-1860) which is seems to be quite forthright in presenting its views in colourful language.

There is a second article about the above in which he is called a 'sawbones' and states that he was fond of cribbage which was how the debt arose.
The Sydney Morning Herald has a somewhat more temperate account of the proceedings, in which he is referred to as a surgeon.
I could not see any further mention of an actual trial.
He certainly is referred to often in other, later Family Notices as "Dr" and appears to be a pillar of respectability.
Judith