Jeff, it is true that, as you say, it did not make any difference. However reasonings like this help to put another brick into our edifice of understanding of what was going on then, and may be useful to keep in mind, later, ... or not...lol.
In that vein, an argument can be made that Fred did know it was Gardner - he was a longtime seaman (sea-captain?), and would know and remember details about recent maritime disasters, much more than the general public, just as people in the aviation community today remember recent air disaster details more than the general public. Anyway, you do seem to agree that he would at least remember that the wreck had happened in the P Islands, and so they knew they were in the P Islands somewhere.
On the other hand, if he knew they were on Gardner, then why did they not just transmit "Gardner" over and over again, for all they were worth? I am not an expert on the post-crash transmissions, but that seems to have not happened. Yes, Betty records 'Ny City' (which is surely meant to have been Norwich City), but that is just part of a larger transmission of other things. I would transmit just one word over and over again - "Gardner", especially given they were newbies in morse keying, and keying off of a mic.
Anyway, it was a thought experiment on my part, and I am sure it has been discussed in some forum here sometime. At this time it does not seem particularly significant.
To the post on Fred passing notes on a bamboo pole to AE as their communication method, while they were flying, that is also my understanding.