My issue is resolved. Much thanks to Apple for helping me out.
First of all, terminology is key. You cannot gift your balance to someone else. However, you can put an UNUSED balance back to the original gift card. In contacting Apple for help, make sure you use the proper terminology for your issue.
To put it another way, if you redeem a gift card but don't use any of the money on it, you can have the full original balance put back on the gift card. At that point, give the card to someone else or redeem it on another account. If you're simply trying to gift or transfer a balance, you're not going to be able to do that.