Where did you get the names of the two American ships mentioned in your first post? You say your man’s name doesn’t appear on either of them.
This Wikipedia article mentions a bit about the history of whaling in WA:
https://en.m.wikipedia.org/wiki/Whaling_in_Western_Australia
A few companies are named. It might be worth chasing up to see if any records survive.
I got names of ships because someone wrote a thesis a couple of years ago which listed whaling ships that had been in Western Australia. Originally, my sister found this document as the Biographical Index of whalers i posted a link to earlier. However, further googling turned up
The full thesis, with a database showing origin ports and number of ships by years. I highlighted the ports with more than 10 ships in the area for the those years and then process if elimination, started with the Port with the most ships. Maths is not my string point but i had to start somewhere so i went with the port with the most ships.
Then, that thesis lists the names off the ships and what years they were here. So i highlighted ships from that port that were here in the right years (i went with a 20 year gap).
Then, i’ve started checking crew lists for names similar to Mullaney. I’m only up to D in the list of ship names but i have found a few potential men, one more likely than others and so strangely it was a ship i had randomly put my finger on previously because it was here at the right time, not because i had seen the crew list. I still have a lot to go through and i haven’t even started on the other ports yet.