Timothy,
Sorry, I know you're techie but I obviously presumed too much PAW specifics. PAW maintains a text based "track" file for each flight containing useful information about, well, your track but also the traffic information sent to your downstream device, and other internal status information. Unfortunatey the format is not public but it's mostly obvious, although some records need a bit of "interpretation". You can download the file from the PAW to your tablet/phone but it sounds like this would need to await your next visit to the aircraft.
Hopefully you have used the PAW web interface on 192.168.1.1? one of the many items is (from memory!) TRACKS, displaying a date and time ordered list of files. Hit "Download" to copy any of them interest to your device. I'm not an Apple user so I'll need to leave you to figure out where they get stored and how to email them to yourself.
One official use of the files is via the flight replay app:
https://playback.pilotaware.com/playback/Upload the track file and you can then drag the little white circle in the lower part of the display to replay your flight. You will see your own aircraft and also all the traffic that was reported during the flight. Hover or click on the traffic to see it's identity. If you can recall a situation of interest (and your posted screenshots would be examples) it would be interesting to see if the ID displayed corresponds.
Alternatively, and perhaps more interestingly if you're that way inclined, viewing the track file in a text editor will show the detail of what was sent to your device. Admin previously described the $PFLAA messages that convey traffic information, and the differing formats with and without callsign. I expect that this will simply confirm that PAW is at times just omitting the callsign but who knows? There may be some pattern or other clue.
About 5 seconds or so prior to the first $PFLAA message for a particular aircraft you should see a lengthy $PALOG message containing the it's hex code and which appears to convey (not very obviously) the source of the traffic. I assume this message indicates the first receipt of information about that aircraft (and BTW why the 5 second delay in informing the downstream device)? I wonder if this message might reveal anything.
Might be easiest just to send me the file? I'm not saying any of this will help, so don't go out of your way to retrieve it, but sometimes an indirect route leads to useful clues and as you say things have ground to a halt otherwise.
Alan (no connection to PAW, just a mis-spent youth).
PS should just add that I have a PAW Classic, not an FX but I hope and believe that the track logging functionality has not changed.