Actually VPN would work fine in that scenario, because you'll be coming at those destination IPs from a different direction.
To try to put it in a more approachable analogy (albeit vastly oversimplified and not 100% accurate to how things work, but let's go with it for the sake of explanation)... let's pretend the packets are packages, and they're being driven from your house to somewhere, or back to your house from there, by a UPS delivery person. If everything is going right, UPS picks up the packages from your porch and trundles off with them, then returns with packages for you.
Now let's say that your neighbor has a delivery box on their porch as well, but it's a small one, and so they send a note to UPS saying "Hey, any packages larger than this can't fit in my porch box, please don't deliver them." But they mistype your address instead of theirs, causing only packages smaller than a certain size to be delivered to you. Now you still receive packages, but... sometimes they don't show up? This is weird, and you call UPS, but all they can tell you is that they're following delivery instructions. Does that mean the person sending you stuff wrote the address wrong? It certainly seems like it could be...
Getting a packet from point A to point B on the internet involves a lot of weird logic, including what packets will be let through or dropped and why (curse you, Maximum Transmission Size!), and sometimes rules get misconfigured. The TCP FIN could be from anything between you (point A) and your particular FFXIV login endpoint (point B). The FFXIV servers could be dropping the connection, your router could be misconfigured, etc. And since the endpoints differ, it's quite possible that you're tripping a broken rule somewhere along the path where your wife's login is not. This could be causing packets not to come back to you because something along the way went "I dunno what to do with this" and cuts the connection off, or it could be cutting you off from the server in certain situations, causing the servers to believe you disconnected and send TCP FIN just in case something along the way needs it.
And the way some of those rules are set up, it wouldn't necessarily be packet loss (which will involve retransmission of the packet in most scenarios) but something along the way deciding that entire connection is bad and arbitrarily generating the packet that says to close it. In this particular case, since the login servers are sending you to one IP address and your wife to a different one, there might be a rule out there somewhere misconfigured which involves the specific server address you are connected to.
That's the reason for the VPN. In our UPS analogy, it's like you go, "Okay, something's wrong here. Let's try getting packages delivered somewhere else" and went to go get a P.O. Box and try having the packages shipped there. In the analogy above, this works; the mis-typed address in the delivery instructions no longer applies to you. Similarly, since the packets to/from you are wrapped in the VPN layer and then tossed back out off the VPN at a different server, different rules are applying when you use one. The VPN is doing the same thing inasmuch as it lets us try a different path to the servers (different UPS drivers, as it were) so that maybe we don't hit whichever thing is responsible.
As I said, I don't actually necessarily expect a VPN to fix that. But if the issue isn't on the SquareEnix side with the FFXIV servers, it might. And if it doesn't fix it, that's definitely a strong piece of evidence that it's probably an issue with the FFXIV servers themselves. In essence, we're just fiddling with the knobs here and seeing which, if any, have effects on the situation so that we know what things are involved in the issue.
At any rate, a number of VPNs have a one-month free trial, so if you do want to follow that road it should be easy to use one just briefly to see if it changes the situation at all. (Just remember to cancel afterwards or whatever.)
Either way, hopefully someone—whether one of us or one of the SquareEnix folks—can help you track down the cause.