Technically you have up to 14 days to add a vehicle to the MID. Best to get some legal advice and If you can prove you have had the car for less than 14 days then it’s definitely worth taking it to court.
Latest one I applied for took over 4 weeks. If you have green new keepers slip you can send that of with your v62 and don’t need to pay the £25 fee. Tick the box saying you have not received the logbook. Will be quicker than just the v62.