I use "to be" but the more I think abt it the more I think "to be at" should be the one that makes the most sense since we say "we are at a place".
"A fun place to be" would be more common, although you could certainly hear someone say the second version and no one would look at you weird. It depends on the context, though.
"A fun place to be" would be correct generally because "A fun place to be at" is kind of repetitive and unnecessary, specifically because the verb "be" already completes the idea of the phrase. The "at" isn't adding anything to the phrase -- it's just directing back to the object of the sentence (the fun place), repeating to the listener that the place that they're at is fun.
Substituting other verbs for "be" you'll probably find they also complete the idea without the "at" being necessary. A fun place to run at, A fun place to make money at, a fun place to inflate balloons at, a fun place to stop at.
From another angle, "a fun place to be at one with the universe" makes sense because the word "at" is continuing the initial phrase into a new idea (I'd say this is usually the job of the word "at"). a fun place to be at 5 o'clock, a fun place to be at a stand-still, a fun place to be at a concert.
Using the extra "at" is pretty common, though. A version that is pretty common is "Where are you?" vs "Where are you at?". People also say, "Where you at?"