You do not need the eartag of the sire to register a calf. You need the breed - or your best guess at it - of the sire, followed by X as it's a crossbreed, and the eartag of the dam.
The problem you will have is that you cannot register a calf as born on someone else's holding, they have to do that and put their own eartags in its ears, with their
herd mark on them.
If the calf had been born at your holding, you would need to have your own herdmark set up (and no, it is not the same as your sheep flock mark nor pig herd mark
) so that you can order your own eartags and register and tag calves born on your own holding. It can take weeks, even months, to get your herd mark through.
I hope this helps.
I would talk to the vendor first, s/he may have eartags somewhere and be able to do the registration and then the move, which would be the simplest. Otherwise, unless s/he can think of something else, I would ring BCMS for advice.