B.C.
How about the furniture tacks that you hammer in to the solid wood on the back (away from the baby)? Than you can tie the strings to the tack to avoid the strings to cause harm and also keep the bumper pad in place. Later on the tack can also be hammered in further and look like it was meant to be there or removed and a small hole will show (unless you patch it).
Velcro is also a good idea (but will most likely cause some more damage to the vintage crib) and if so you might want to consider the Industrial kind (no little kid will get it off, it causes me problems), but it might take the finish off if/when you want to remove it.