DY: It’s brought down in several Seforim, actually. Basically, one day he showed up to Yeshivah really happy. R’ Nosson Adler asked why. He answered that he saw a guy on the way, realized he was from Zera Amalek, and was Mekayem the Mitzvah of killing him. It’s clearly made up, but my point stands.
MA: Interesting. I wonder if Binyan Beis Hamikdash is Doche Tumah. We could say yes via Kal V’chomer, but there’s got to be an actual Makor one way or another. Tzarich Iyun. Either way, though, that doesn’t make it impossible Bizman Hazeh. That just means that we need a Parah Adumah first (or to raise kids in a way that ensures they never come in contact with a Meis).