So technically, I can't deny this is the right answer. But, still, what the OP is adding is just a laminated card for a TB that's still functioning, so there's no reason to think the TO's released a proxy, and this isn't a knock-off, it's just added documentation in line with the published TB. OK, I still concede that it's not strictly kosher, but, to be honest, I'm sympathetic because I think those cards are a good idea, I'm just too lazy to put one on my TB.
Here, I think the OP has crossed an important line. If the tag "goes missing", then it's gone. In that case, it's just flat out wrong -- wrong! -- for your laminated card to assume the identity of the missing TB that's the official owner of that code. So your added card definitely should not have the secret code on it. I understand your thinking, and it's handy to have the code on the card, but only the TO has the authority to put that code on anything.