Hi,
I am not home, could be several weeks before I get back home, but some thoughts ...
marrying 1837, yes, possibly the clergy was on a circuit, but yes, he was expected by the NSW civil admin (so still a penal colony, no representative parliament etc), to transmit to his denomination's HQ ... (Diocese) the details of baptisms, burials, weddings of ceremonies he conducted , transmitting four times per year minimum (Jan, April, July, Oct)
also possible to transmit more frequently, and of course if on a circuit, to errr ... delay or misplace the register in the saddle bag ...
So not all records are extant, BUT ...
Marrying 1837 ... and Alice likely born 1844 ... so not the first born ....
You likely have name of the clergy conducting the ceremony in the Maitland district ... thinking about the location for his register that he kept with him .... and from which he copied his records for transmitting to St Marys Cathedral ...
Think about any possible older siblings for Alice ... are their baptisms found? who was clergyman ... same or different from the marriage ... where was the baptism ... part of the Maitland circuit or in a different district ...
So basically trying to form a view of where Alice would likely have been taken for her baptism ... and then try to figure out who was likely clergyman ....
Sorry, I am not able to get into the actual nitty gritty search...
PS, I am assuming you have gone right with fine tooth comb : familysearch with Australia as location; Trove; NSW BDM; NSW ARK KIT ... perhaps available Qld State Library or larger public libraries or perhaps you could contact Ask a Librarian ...
https://www.records.nsw.gov.au/archives/collections-and-research/guides-and-indexes/archives-resources-kit-ark (for the printed Guides ... no longer used by NSW Archives, but very informative for background, and gives clues for suggested thingys !)
Fingers crossed
JM