There are a few ways to get around the bug.
1. Destroy the relationship with mean interactions so that you are no longer BFF, and then get a new BFF.
I actually didn't think it was a bug. I have accomplished this aspiration with 7 of my sims, using the way mentioned above. It is a little frustrating, but, then again, in real life, it's hard to maintain more than one BFF... and I do like that The Sims 4 feels somewhat more "real to life" in the social aspects that Sims 3 did.
It is also good to know that some neighborhood sims already have BFFs, and you have no way of knowing until you get to the point in the friendship in which you "should" be able to become BFFS, and, even though your sim does not have a BFF, you cannot become BFFs with that other sim. I started making a list of them for my game actually since I got tired of forming close friendships with sims only to be denied the opportunity to makes them BFFs. For example, neither Bella nor Mortimor Goth can become best friends with any of my sims (at least in my game). They are an unplayed household for me, and I assume they are already BFFs with each other. For that reason, I stopped trying to become BFFs with any married sims. Another sim I can't become BFFs with is Hana Shan. She's also married. After discovering that, I started becoming BFFs with children mostly. They seem easier to befriend in general. Of course, I do feel mad having to "Yell At," "Insult," and "Scold" them over and over to end the BFF status each time.