I pulled up my Parenting Plan (every state is different, but the verbiage might be somewhat similar). Mine says that both parents are entitled to "the right to unimpeded telephone conversations with the child at least twice a week at reasonable times and for reasonable durations."
I also don't get why your ex wants to exclusively facetime, other than he's nosy about your home?
Anyways, as long as you're not impeding your children from talking to him, you're fine. He's trying to manipulate their mindset with that "why won't mommy let you talk to me?" comment. Start journaling. It's possible he wants it down the line where you have 50/50 custody or something where he doesn't have to pay you child support. My ex started gunning for that after he got remarried. I suspected my ex's new wife didn't like him having to write me a CS check every month.