Unfortunately the data provided by Wireshark does not proof that the server closed the TCP connection to your PC. A packet with a TCP FIN flag could have come from:

a) the application running on the server, if it actually closed the connection (or crashed and the OS closed the connection)
b) any network node on the route between your PC and the server’s location (including your router)
c) software running amok (brain-dead “anti-virus software”, firewalls, ...)

AFAIK there’s no sure way to figure out the source of a FIN. You could take a look at the TTL (Time To Live) field in the IP header of the last few packets received from the server prior to the FIN and the FIN packet itself. If the FIN actually came from the server, the TTL values should be identical. If the TTL of the FIN packet is different, it probably did not come from the server.

You already tried a few things with your wife’s Windows user and PC. You still get the disconnects if you use her Windows user on your PC, but with her FF XIV character everything is fine? If you use her Windows user on her PC, you also get the disconnects? If the only difference between a stable and a bad connection is the character being used, I don’t see how this could be anything on your end...

Did you try creating a fresh character (just for testing purposes)? That could clarify whether it’s somehow related to your character or your Square Enix account.