petthefabric

I think the interfacing would work. I'd cut it wide enough to stabilize the seam.

Another option is to measure the border, pin it to the seam, steam press the bias edge to fit the border. Then stitch with the border fabric on top. This may also be a time for hand basting-extremely rarely used.

Just thought of something else. The cross cut of fabric (perpendicular to the selvage) is a little stretchy. So if the borders are cut that way, the stretch can be used to advantage to fit the stretchy bias. It'll go back to it's normal length. This helps to avoid puckers.