BPP version of a problem related to #P completeness

Given a $$CIRCUITSAT$$ instance $$\varphi(n)$$ in $$n$$ variables and a fixed $$k>1$$ the problem of deciding if the number of satisfying witnesses is $$2^n\big(1-\frac1k\big)$$ or $$\frac{2^n}k$$ is $$PP$$ complete.

Is there a problem related to the above which is in $$BPP$$?

• For fixed constant k > 2 this a Promise-BPP problem. Pretty doubtful it's PP complete. – Ryan Williams Jul 22 at 2:43