Hi there,
Back in 1810, the incoming governor (Lachlan Macquarie) overhauled the penal colony's admin system and basically - perhaps following his own Scottish tradition - the recording of the names of females from at least 1810 until perhaps the end of the penal period (around 1840), meant that on official records whatever the SURNAME the female arrived with, STUCK with them, regardless of how many times they married/co-habited/had children by various domestic partners. The women tended to use their husband's surnames for everyday matters, but the Admin bods tended to keep the surname the womenfolk were recorded on arrival. Consequentally for example, at least three of my (g x ?) grandmothers/great-aunts are on the 1828 census under their childhood surnames (one under their unmarried mum's birth surname), despite growing up in Sydney Town, marrying in Sydney Town, and having babies - in Sydney Town ! and ... wait for it .... still living with their 'lawful' husbands in Sydney Town - all before the 1828 census was announced.
The Britomart is listed at the Colonial Secretary Index at the following link :
https://colsec.records.nsw.gov.au/ and
Daniel PEACHE was the Master of the vessel at that time, and he is also listed there.
A lass named Catherine DUNN is also listed there, she is seeking permission to marry at Liverpool, May 3, 6 of 1822. Reel 6009 etc....
My offline resources give me further information about the marriage you mention. She was NOT married at St Mary's. The clergy who conducted the ceremony at Liverpool has obeyed the church requirements and the civil requirements and the information has been received by the Roman Catholic authorities and is recorded in the Diocese records held at St Marys Cathedral. Roman Catholic ceremonies were not formally recognised by the civil admin system until around 1825, so it is fortunate that the details of the 1822 ceremony were transmitted through and that there is an extant record.
Here is some information from my notes re the likely marriage you are writing about. (I have a research interest in NSW marriages in those early years).
NOTICE there's a lack of a clergy name, and that there's NO signatures (not even X mark), no occupations, no abodes, no ship of arrival, no ages, no parents names ...etc.
20 June 1822
Michael BYRNE and Catherine DUNN
Witnesses (THREE) : Farell CUFF, James DEMPSEY, Mary DUNN.
Likely you will also find the marriage listed at the on-going project, Biographical Database of Australia. There is a nominal annual subscription fee for that website :
https://www.bda-online.org.au/who was Mary DUNN? JM