I’m a bit perplexed as to why anybody in a powerful position would talk to anybody with press credentials. It’s clearly not a very intelligent thing to do... That being said, if he did indeed tape the conversations, then he obviously informed people they were being taped. Otherwise he would be breaking the law. So release the tapes and prove the conversations were real. Otherwise, it’s just here say.
No, he doesn’t have to in DC. It’s one party consent:
https://www.rcfp.org/reporters-recording-guide/state-state-guide/district-columbia