If I recall correctly he said he had Michonne back at Woodbury as a hostage (which I did forget) but that still doesn't make sense to me. So lets say he DID have Michonne, the group can't assume he was bluffing but they also had no proof she was even alive, and they had no way to save her if she WAS alive. They DID have proof that Tyrese was alive (he was right in front of them alive and healthy) and had they killed the Governor right there that's at least 1 life that's GUARANTEED to be saved. Just doing the math of the situation it's not feasible to trade the potential to keep one person alive (that you really had no way of saving even if he wasn't bluffing) by letting someone you can DEFINITELY save die horribly. They traded and ACTUAL life for the promise that someone else has a CHANCE of surviving... maybe I'm wrong here but I think the better option is to just kill the Governor, save Tyrese and hope that the war dies with Philip then worry about saving Michonne later, if she's even alive and if he wasn't bluffing about having her.
I can't post pictures of the climatic scene but the Governor said that "they had Michonne." I always assumed this meant she was in the truck but she could be in the camp I guess
If the group did assume Michonne was on the truck, they said that they just had to hold out and hope that Michonne would "have enough time to escape"
Tyreese wasn't guaranteed to be saved. The Governor had men with him. He was badly beaten and bound. If Billy shot the Governor, Tyreese could've tried to run but then just get mown down by the Governor's men and then Michonne would be guaranteed to die (once the Governor arrived back at Woodbury)
Plus, Billy DID shoot at the Governor. He missed. Andrea had already left the prison at this point so she couldn't have done anything
So in summary, Tyreese is guaranteed to die no matter what they do. If they try to kill the Governor, Michonne is definitely going to die and they want to give her a chance (Rick did at least).
The best possible situation would be the group knowing the Governor is lying, killing him, and also dooming Tyreese in the process. You are correct that they should've shot and killed him (but Andrea wasn't there so if they missed the shot, and they did, Michonne would be doomed), but with the knowledge that they currently had, they made the right choice. It was logical to assume the Governor did have Michonne if he had her katana
Imagine a group of a hundred motorcycles driving down a freeway. Eventually, they hit a junction. One road goes northwest and the other goes northeast. So one guy, we'll call him S, says, "Let's go northwest!" A mile past the intersection, a semi careens into the group and kills ninety of them. Ten are wounded, but they survive and keep going. Eventually, they hit 10,000 miles. S suddenly has his consciousness thrown into his past body right before the junction. Now, he says, "Let's go northeast!" All 100 bikers survive. Happily ever after, right? But what about the ten, no nine, who went northwest and survived? What happens to the reality they were living? Does it just disappear now that S has changed the past? It's not like only bad things happened on that 10,000 mile journey. Maybe one of them fell in love with a gas station attendant and got her pregnant or maybe one adopted a homeless kid that joined the adventure. That 10,000 mile journey would be full of stories. Romances, farewells, friendships...the loss of those ninety lives is horrible and unfortunate, but what would rewriting their history mean? The nine who survived lived full lives and did the best they could with the hand they were dealt. How could it be right to just erase all that? Isn't that worth something? Is there a point to a world where everything is happy? Are people who struggle for a better life just idiots? Being human is about fighting even when it seems hopeless and finding happiness in a world that hates it. Are you saying that's worthless?