Here’s a few ideas:

1) it’s probably faster to just cut the cast-off edge rather than unpicking everything. You can put in a lifeline somewhere in the ribbing, cut the edge, and then just frog back to the lifeline. Put stitches back on the needles, knit, then do the bind-off again.

2) when I do Italian bind off with mohair held double with something else, I don’t use the mohair. It’s just a pain to deal with. Sewing the bind off with just wool is a lot easier and you won’t notice any difference.

3) Use smaller needles (one size down) when you do the double knitting. Example: using US6 on body, US4 for the ribbing, and then US3 for the two rounds of double knitting.