Turns out it was just the short circuit code that needed to be cleared. The FRM was still in working order.
From what I've gathered after looking into all this, since I only had the 1 malfunctioning bulb and nothing else that the FRM controls (mirrors, windows, and other lighting) was malfunctioning the likelihood the FRM was fried was low. Had there been 2 or more things malfunctioning it would have probably been a good indicator that the FRM was a goner and needed to be replaced.
Had quite the argument with the dealer, though, trying to get them to reset it. They were quite insistant that I didn't know what I was talking about and that the only option was to replace the FRM. After standing my ground and demanding they actually listen to me, they reluctantly took it in back. Over an hour (and $62) later, turned out I was absolutely right.
I'm quite upset it took them over an hour to do something that shouldn't have taken more than 10 minutes and still pretty put off that they wouldn't listen to me in the first place. It goes without saying, but I probably won't be taking my car back to that dealer.
To think, BMW wanted $250 to replace the bulb originally and then would have wanted another $500+ to replace the FRM module. $750+ in total charges avoided by a simple reset instead.