Did the car have a number plate? if so they registered it :O DVLA do need specifics to give a registration over and the most important one of those is the vin! on your sales sheet was there a VIN number with registration, if so it was registered as the dealers will only get that reg with said VIN.
When you checked the reg was it on dvla website? if not then details wont show up on other pages anyway from anything to 2 weeks to 2 months.
On a second note I would of took the complaint to seat hq not motability, on dealers using trade plates, under new laws unless the car is a demo registered to them they have to use trade plates now, even if the car is registered if its not a car registered to them.
But still no harm no foul, they have give you a replacement 😉