If they killed Pettigrew, they could simply claim that he died in the course of his capture. They don't need him to be alive to prove that Sirius was innocent; they just need his body.

Pettigrew was a Death Eater, who had successfully pinned one of his previous crimes on someone else, and who had managed to evade detection and capture for over a decade. He's clearly a dangerous and ruthless individual who is capable of taking extreme acts to remain free (like pretending to be a pet rat for years at a time). No one would blink an eye if you failed to bring him in alive.

Of course, telling that lie would require Lupin and Black to be less than perfectly honorable and scrupulous - but they weren't saints.

He wanted revenge...

...for Jame's and Lily's deaths. After all, it was Pettigrew that caused them to die!

“Harry, this piece of vermin is the reason you have no parents,” Black snarled. “This cringing bit of filth would have seen you die too, without turning a hair. You heard him. His own stinking skin meant more to him than your whole family.”

As for the Ministry realizing that they were murders, everybody thinks Pettigrew is dead, anyway! They would never realize that he had actually been murdered!

“When I cornered him, he yelled for the whole street to hear that I’d betrayed Lily and James. Then, before I could curse him, he blew apart the street with the wand behind his back, killed everyone within twenty feet of himself — and sped down into the sewer with the other rats...”

After they've killed him, it's easy to lie about how he died. Perhaps the all-consuming guilt forced Peter to put a wand to his head and say Avada Kedavra! After being confronted by Lupin and Sirius, he began to finally feel remorse for his actions.

^{(There are possibly better lies out there! I'm not great at lying, myself!)}

So, really, they have nothing to lose by killing him, they would get their revenge, and nobody would find out! Brilliant!

This will not be strictly illegal

1. Peter Pettigrew is a death eater. Aurors and other members have killed death eaters in the series. It is mentioned that Barty Crouch authorized to kill many death eaters during the rise of Voldemort. Mrs.Weasly was not arrested for killing Bellatrix.


2. They don't have to use the unforgivable curses. Harry nearly killed Malfoy without using one. Sirius and Lupin are skilled wizards who can kill a person without using an unforgivable curse.

Besides, they could claim that it was an act of self-defence or they did it to defend Harry.

Lupin and Sirius were after justice and revenge. Neither of them had anything to lose and even if they had, they couldn't see past their hatred for Peter's betrayal of their best friend, to consider their own lives and the outcome.

Lupin, was a werewolf and his kind were not treated well by the Wizarding community. Most of them couldn't get jobs, and lived underground. He finally had friends at Hogwarts, best friends that meant so much to him, as well as how much Lily meant to him, and lost everyone.

"For the first time ever, I had friends, three great friends... I was terrified they would desert me the moment they found out what I was. But of course, they, like you, Hermione, worked out the truth... and they didn't desert me at all. Instead, they did something for me that would make my transformations not only bearable, but the best times of my life. They became Animagi.">

"Your mother was there for me at a time when no one else was."

Lupin goes to great lengths to emphasize how important his friends were to him. This makes the betrayal of one of his friends (Peter), and the recent realization that this whole time, one of his friends never betrayed James (Sirius) so much more powerful. He would not abandon him or try to stop Sirius after finding out the truth. He also wants justice for the betrayal.

‘You should have realised,’ said Lupin quietly. ‘If Voldemort didn’t kill you, we would. Goodbye, Peter.’

I think it was Sirius who wanted to kill him first.

“They’ve — got — a — right — to — know — everything!” Lupin panted, still trying to restrain
Black. “Ron’s kept him as a pet! There are parts of it even I don’t understand, and Harry — you
owe Harry the truth, Sirius!” Black stopped struggling, though his hollowed eyes were still fixed on Scabbers, who was
clamped tightly under Ron’s bitten, scratched, and bleeding hands.
“All right, then,” Black said, without taking his eyes off the rat. “Tell them whatever you like.
But make it quick, Remus. I want to commit the murder I was imprisoned for…”

And Remus was the type of person who followed his friends plans, I mean he was like this since he was a student, he didn't stop James and Sirius when they attacked Snape although he was a prefect and he came along them while breaking the rules. Creating the marauders map, accompanying them while they became unregistered animagus, and we don't know what else.

“I sometimes felt guilty about betraying Dumbledore’s trust, of course… he had admitted me to
Hogwarts when no other headmaster would have done so, and he had no idea I was breaking the
rules he had set down for my own and others’ safety. He never knew I had led three fellow
students into becoming Animagi illegally. But I always managed to forget my guilty feelings
every time we sat down to plan our next month’s adventure. And I haven’t changed…”

Perhaps he kinda felt like he always owed them, because no one else would have stayed his friend after learning his secret, let alone what they've done to make him feel better.
Also he may have felt guilt because he was the reason Pettigrew became close to Potters and knew their place.

Remus, always the underdog’s friend, was kind to short and rather slow Peter Pettigrew, a fellow Gryffindor, whom James and Sirius might not have thought worthy of their attention without Remus’s persuasion. Soon, these four became inseparable.

_{-- Pottermore}