Is it possible to play XIV on integrated? Absolutely. People don't give integrated as much credit as it deserves these days, assuming it's modern parts (primarily the CPU). It's nowhere near as bad as it was 10 or 20 years ago. Just don't play it on high settings. As with even dedicated cards, the lower your settings, the smoother it should be, if you have graphical lag.
From here on, I'm going to assume what you said is true, about implying you're not asking for advice on the GPU for anything but FFXIV.
As for the GPU purchase, Tom's Hardware tends to have a decent hierarchy listing of GPU. HERE
So there's some unanswered questions that you need to consider. Are you stretched for cash or do you try to minimize expense as much as possible? As in, maybe $60 or so saved is worth a small drop in potential power. If yes, then the RX is going to be your go-to card series. 4GB cards are fine for XIV. Not perfect, but you probably won't care much if you're gaming on a low budget.
Do you have a preference between NVIDIA or Radeon? IMO, that's pretty much the deciding factor on the matter, if you do. As others noted, the GTX1060 is your ideal budget "powerhouse" today. That's assuming you prefer NVIDIA and vice versa. If it doesn't matter on brand, then cost point will be slightly cheaper for the RX480, as others stated. Assuming saving on cost is a high priority though, the RX470 is capable enough and a little cheaper.
While both card brands work fine regardless of CPU, they do tend to behave slightly better with their respective/associated brand, in my own personal experience. NVIDIA with Intel, AMD Radeon with... AMD. Many don't notice or experience the problems I've seen, and they've mostly been minor annoyances, so it's not a huge deal, but could be something to consider. I personally wouldn't worry about it though, at least not over cost or brand preference.
TL;DR - The RX470 or 480 is probably your best bet to budget with, assuming you're running good parts elsewhere (e.g. CPU and RAM). Given the small cost difference of ~$20, the RX480 is probably your better option.
This YouTube link will show you the benchmark of the RX470 and HW running at max settings 1080 (their channel shows a ton of benchmarks for a variety of cards, if you're curious): HERE