No, it would not work, as the serial number would not be a BT YouView box. The serial number is used for licencing and copyright validation.
So I do think that as along as you have a subscription to Bt TV the box doesn't matter.
That's not true and you should be careful not to post inaccurate information which might lead to others to waste money.
Your TalkTalk box worked because it's a YouView box and all YouView boxes can be used BT, TalkTalk or any other ISP. But the Panasonic box is NOT a YouView box so it will not work with BTTV. At all.