As far as I aware there is no bounded probability distribution where the mean and variance are independent of each other, and independent of the distribution parameterization. I don't believe it is mathematically possible. The bounds impose constraints the variance and, if the distribution is not symmetric about the mean, they also influence the mean.
Perhaps if you give us some more information about your application, we might be able to help with some approximation that fits your use. That is, something similar to James Tursa's answer and comment above.