Why did the Chechens leave the money in the bag with the bloody arm?

The prevalent atmosphere when the Chechens visit the bar has usually been one of tension and mistrust. They don't entirely know if Bob and Marv might not ultimately have a hand in the robbing themselves and being the mistrustful gangsters they are, they make it pretty clear that they're the bosses and want their money back!

We even learn at the end from Chovka that he knew Bob was responsible for the robberies. And while it's not clear if they already knew it at that point and just went on with the game to see what happens or if they only learned later, at best they were mistrusting them.

So giving them the money after finding and killing one of the robbers achieved multiple things:

• It made it pretty clear that they're not fooling around (especially when adding the severed arm with the wristwatch that Bob told them, and the cops, about). It's a matter of intimidation and showing them who's in control (and that it's them and not the cops who found the wristwatch).
• Besides that, it also served as somewhat of a test. Now even if they had a hand in the robbery it would be pretty stupid not to give the money back, but even then it would still be interesting to see how they react to the death of their supposed partner.
• And it's also somewhat of a little punishment for them letting themselves get robbed, by having them deal with some of the mess of the aftermath, too. Afterall, it's Bob who has to get rid of that arm and Marv to clean the money for them.

If they're innocent, it shows them "that's how we deal with stuff like that!". If not, it tells them "we're on your tracks and this will happen to you too!", and in any case "go clean up your mess!".