I don't think it's BattleEye, I've also been getting those messages lately and I play on a PS5.
As much as all the official communication (error messages, FAQs, manuals, etc.) would have you believe the problem is on your end, I'm afraid the reality is that Bungie's server infrastructure is just shitty and overloaded.
When the game says "You are experiencing network difficulties", I read it as a plural you, as in "You the players".