The solicitation to commit treason came to Oleksandr Vilkul on the second day of the war, in a phone call from an old colleague. Mr. Vilkul, the scion of a powerful political family in southeastern Ukraine that was long seen as harboring pro-Russian views, took the call as Russian troops were advancing to within a few miles of his hometown, Kryvyi Rih. “He said, ‘Oleksandr Yurivich, you are looking at the map, you see the situation is predetermined,’” Mr. Vilkul said, recalling the conversation with a fellow minister in a former, pro-Russian Ukrainian government. “Sign an agreement of friendship, cooperation...