I had this same problem, but you might have already tried what I did. I've been an Android user but recently got an ipad and had problems with the facebook login. It was going through safari though. There's been discussion on the twitch live stream this week that they're working on a fix to the issue users are having. But I'll share what worked for me below in case it helps.
What I did was install the facebook app, after unlinking the account. I also had my app connected via google play games and my Android phone, which allowed me to go to facebook settings and delete the account, since it was backed up in several places. On the facebook website on PC, I think it was settings/apps and websites, and I removed the NBA Live Mobile account. Then I installed the facebook app on my ipad, logged in and all that. Reinstalled NBA Live Mobile on the IOS device, and went into the NBA Live Mobile settings and linked the account again, allowing the permissions that it asked for. After that, I started seeing my NBALM username on facebook when I opened the app before logging in. That also resolved the playing with friends feature that wasn't working prior to installing the facebook app on IOS.
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