I'm not sure whether there is a way. Unless something like WebElement.super returns the parent driver object from which the current element is derived.
However, do you have multiple WebDriver objects? I there's only one, then just make it a global variable. Or set a global variable that holds the current WebDriver object in use.
Or, you could pass the particular WebDriver object you need in the same set of parameters when passing the element to the other class, which would at least be consistent with the passing of the element.