I'd like to try and tackle this if I may although not via Rouche': First, it's easy to solve for the roots of in terms of the Lambert-W function:
and keep in mind the W-function is infinitely valued and looks a bit like the log multi-sheet so we should actually write it as:
where is the sheet number. The top left plot is the contour for and the zeros as the black dots. As you can see, the contour is enclosing three zeros. The lower-left is the image of over that contour. On the brown contour we have: and if is large, the expression is dominated by giving rise to the circular contours around the origin but note the remaining contours complete the circuit around the origin giving rise to a winding number of . This is primarily due I think to the blue contour over which we have and since the term is small, the real part is dominated by and if this is positive, then the circular contour complete the circuit around the origin times as shown by the lower left plot. However, in the case of and as the second set shows, then which causes a detour around the origin and gives rise to a winding number of . This is reflected in the top right plot showing only two zeros enclosed by the contour . . . I'm thinkin' B at best.