I agree to stay stitch that edge you are going to join to the border. Use a smaller stich 2.5 or 2.0 . Starch the edges before stay stitching.
But it could be very problematic if you use cross cut fabric rather then lenghtwise grain for your border. Lenght wise does not stretch .. but cross cut ( across the width of fabric) will ... so combining a cross cut and a bias could give a really off kilter result.... in other words you could have the whole outer perimeter stretched bigger than the rest if the quilt.