Resolved. I received an email this morning once my game time expired informing me that my sub wasn't paid and my service was cancelled as a result. I was then able to access the Mog Station without any difficulties, and paid my sub. Sadly I don't know what caused my payment to fail in the first place, but considering there's two points of failure here, I'm assuming that either the account management system or Steam had a bit of a hiccup, possibly due to the new year (that seems to be causing a lot of tech problems for some reason).

If you found this thread via Google or whatever search engine and you're in similar circumstances, don't panic. While I recommend submitting a support ticket to Square and Steam, I am rather confident in saying everything will be OK tomorrow. You'll just need to setup your auto-payment again. If you're worried about your veteran rewards, I can personally say I didn't lose any progress towards mine.