When I have had problems like that I've had good results looking up the name of the ship and checking the passenger list. Might have to check several voyages and years. I don't know if a convict ship would work the same way but it's worth a shot. I just checked it out on Google and find lots of references to that ship so all you need is the date approximately of their arrival in Australia and their name to look up would presumably be Lewis. I'd start looking about 1800 for somebody Lewis and wife.
When u come up with a couple with that name more research w/need to be done to see if they match up with the birth of Edward in 1804. It's a lot of work but I have done it with a similar problem. (You say it has been suggested the parents came on Neptune - check further into that source too; see if you can find proof in family records or whomever told you that info). I've had to do that too and none of it is easy.
But you have problems with your dates as there are not 35 days in a month and would he have married so late in life and long after his child was born? Or it could be that he was ill and so before he died he wanted to do the right thing and marry his child's mother.