FAR 61. Whatever says a DPE must hold at least a third class. On the FAA web site in the Basic Med FAQ section there is a statement to that effect. So a friend who wished to go Basic Med and continue being a DPE filed a petition for relief from that requirement stating that it was probably an oversight and it makes sense that as long as the DPE was in an airplane covered by the Basic Med rule, the DPE should be good to go.
He he got a letter back saying there would be no action on his petition because he did not require any relief because certainly, a DPE could operate under Basic Med. I have a copy of that letter but not sure I can attach it but if somebody wants it in email, pm me.
My friend just wrote to Flight Standards showing them the letter he got from "the other FAA" asking them to correct their position.