Nothing about Butchers blackmail statement is unbelievable to Edgar.
Vought is dealing with Homelander killing an innocent person and the country is pissed.
Vought is dealing with Compound V being exposed and the country is pissed.
Vought would also then be dealing with the face of the company not only being a murderer but a rapist. Vought also knows Homelander is extremely unstable and dealing with the rape story in addition to the murder could throw him over the edge (they literally have a scene displaying Homelander is on the brink of just destroying everybody). That could mean the end of Vought altogether.
Vought is more important than Butchers life. Everyone understands that, even Butcher. Thats why he had the upper hand and Edgar let him go. Bro, there are people who thought it was worth dying to have their children have the opportunity to vote. Thats fact. So yeah, its entirely believable there are people who would trade dying for getting rid of Vought.