I have implemented Attention Points at one of our clients.
One of the things it creates is an internal "Supply & Demand" of attention (or in simple words time). For which the Points are its currency to exchange attention.
I could definitely see a similar way to do budgets. You allocate "Budget points" (you might just call them Euro's) circles and roles. Which is for them just like attention points a resource to spend within the role.
Its definitely one way to do budgetting. It is kind of a cheque for each circle/ role to spend on what they find important for their role up to their max "Budget points".
You could then define some auxiliary processes/policies that add a bit of alignment between points spenders if you need for larger amounts of descretionary spending.
Hope this helps,