We would be able to help you more if you could post the sketch of your bag.
If the bag has a single long strap, it will be hard to turn and get a clean edge. A shorter strap will not need a stay.
I suggest that the width of your straps should at least be the same as the width of your foot. It will keep the foot at an even level as you top stitch.
Keep in mind on how you can reduce bulk especially in the areas where it loops around d-rings or buckles or even if the ends of the strap are inserted into the seams.
This is how I dealt with the bulk of the wristlet strap that I made out of pig suede. I clipped the corners.
It is only 14″ long by 1 1/8″ wide. The finished width is 5/8″ wide. The short ends have a 3/8″ seam allowance. So once everything has been turned and folded, the ends are not as thick:
I hope that it will help you move forward.