As tensions flared over disputed territory in the Caucasus region in the summer of 2020, Azerbaijan’s squadron of high-priced Washington lobbyists scrambled to pin the blame on neighboring Armenia and highlight its connections to Russia. Unbeknown to members of Congress, Azerbaijan had an inside man who was working closely with the Azerbaijani ambassador to Washington at the time on a parallel line of attack, according to text messages released by federal prosecutors. Representative Henry Cuellar, a Texas Democrat now charged with accepting bribes and acting as a foreign agent in a yearslong scheme, indicated in a text that he planned a legislative maneuver to try to strip funding from Armenia because it hosted Russian military bases. Azerbaijan’s ambassador responded enthusiastically. “Your amendment is more timely than ever,” the ambassador, Elin Suleymanov, wrote to Mr. Cuellar. “It is all about Russian presence there,” added Mr. Suleymanov, who referred to the congressman as “Boss.” Mr. Cuellar’s legislative gambit did not go far. But by the time of the text exchange, his family had accepted at least $360,000 …