How can a Federal judge issue "sanctions" against Trump in a civil lawsuit?

I suspect that the BBC have realised that they may well be playing from a position of strength here. They can tie the case up and present Trump’s side with only unpalatable options.
Taking on the BBC isn’t something I would be doing. A loss is near impossible to enforce. Maybe demand a slice of the BBC international sales in the US from now until the Sun goes cold. It is unlikely to be more.

The BBC is often accused of a left bias, which despite protestations, has some truth. The danger for Trump is that some part of the BBC hierarchy starts to regard this as a sport - Trump baiting.

Demanding much the same amount of money as the entire HS2 rail project is not going to win him any political leverage over the BBC.

I would guess, though, that the threat of a seizure of the American IP to its shows and franchises would put quite a pressure on the BBC to comply with an American judgment.