So long as your new property can get FTTC (not every line on an enabled exchange can get it) then yes you can stay on the trial. When you move you'd take the router we supplied with you but leave the BT modem. You'd also need a new appointment arranging for the new property (possibly two engineers if you need a new phone line as well).
We've not done any house move orders yet so would be good to do one. When you're ready to place the request with us I'd suggest calling in to make the request and ask whoever you speak to to speak to the fibre team or me so we can arrange it.
Enterprise Architect - Network & OSS