Well, update, thought it's not really a positive one, and seems to only further confirm some of my suspicions here.
I was at a buddy's house last night, and brought my laptop (a different computer than the one I normally use). I installed BF1 via Steam, and sure enough, same issue.
So, to summarize; different PC, different game installation, different wifi network, but the issue persists, same as before. For me, this fully confirms that for some reason, my account is the issue. Given that EA reports I have no bans or anything, it's very frustrating to not be able to connect with any support engineers at EA who might theoretically be able to see or fix the issue with my account.
I guess if I really wanted to, I could make a new EA account, buy the game (for the 3rd time at this point!) and I suspect that would work. Unfortunately, having hundreds of hours in the game already, and all the weapons unlocked, the prospect of paying EA money and starting over because they broke and won't fix my account doesn't exactly sit well with me.
Maybe someone from EA will see this and help me figure out how to actually reach a support team capable of understanding or maybe even fixing my issue!
Seems unlikely at this point though :(