Using Persistent Key is poor security practice
That said, when I started streaming earlier this year, BF Persistent Key worked fine. I switched to non-persistent key 4+ months ago, so can't comment other than /Live/Producer page has bugs, and FB has been updating things in background... but if you are setting up FB Live Event correctly with Persistent Key, it should use the same key (though you may need to schedule event, wait some time (hours, day, or??), then check upcoming event and copy the (possibly new) key, then test. Then create another new event, and check if the same key??
If the key is changing, you should probably contact FB for support