I would consider buying the 1/4 inch bias tape all ready done from my local quilt shop. I know they made it for doing stain glass applique a few years ago. If you can't find the colours you want then you could make it. On 1/4 inch wide bias, I would try a sample at 1/2 inch wide. That would give you a 1/8 fold under on both sides. You don't want the fold unders to peek out from the 1/4 inch covering strip.
With narrow bias strips, I like to put the raw edges together and stitch down the 1/4 inch width onto the pencil line on the background square. I then push the 1/4 inch bias strip over the stitching line and hand stitch the remaining side down. This may not work for your design but try a sample and see. I would add no more than 1/4 inch to each side of the 3/8 strip for fold unders (3/8 plus 1/2) or less.