if your batting says quilt every 4" that does mean---in all directions---if you leave your 2 1/2" borders totally not quilted the 2 1/2" part is fine...but what about that length??? is it less than 4"? you need at least a line across every few inches to keep the batting from pulling apart and bunching up in the borders when laundered.
if you do not want to quilt areas of a quilt it is best to use flannel or fleece for a batt- which will not pull apart and does not have any specific quilting requirements

I agree with others...it's your quilt.I vote for a lazy S....either where the border attaches, or down thru the center. Works for me.

No, with a 2 1/2 inch border it is not necessary to quilt. That said, it's you're quilt and after looking at your design already completed you can determine if it is needed to add texture/appeal to the rest of your quilt.