Outside of replacing the binding, your fill method is probably the only way to go. I might be tempted to use a glue that will be more appreciative of your stain than superglue though.
I'd even be tempted to go a filler like Timbermate or Earl's. The repair won't be invisible, but at least the filler will take the stain. CA will not.

If you use a filler, you could either try and match it to the colour of the veneer or even completely contrast it with something like ebony.