I have actually been in live chat with quite a few Advisors over this issue.
Some of them said they added the skin to my account, but once I check it turns out it's still not there. Apparently they were confusing the skin I was talking about (The Care Package) with one that I already unlocked via the Play to Give mission (Package from Home), and so they were adding the Package from Home skin to my account, instead of The Care Package.
Any Advisor who actually understood which skin I was talking about ended up telling me they couldn't grant me the skin, and one Advisor took it up to a Specialist, who then told the Advisor to send me to the forums (hence why I'm here).
I ended up emailing support instead, and the information I got through email was much more useful. They told me that it unfortunately was not possible to grant the skin back for me, but they could give me a compensation battlepack. I was and still am upset over losing the skin, but at least they're kind enough to provide me with a form of compensation. I agreed and also asked them to forward the bug to DICE if possible, but I have still not received a reply since, and haven't received my compensation yet.
I hope I clarified everything. I have already gotten into live chat with support, and emailed them, to no avail.