I had a somewhat similar experience when I bought a 200 in MN and had it shipped to me. They collected all the taxes and fees up front to remit to the DMV, but they messed up every financial aspect of the deal through and through. I needed two 60 day temp tag extensions because of their gross incompetence processing the title, but eventually I got a packet in the mail with a check from the dealership made out to the DMV for taxes and fees (it was literally just my own money they were passing through the dealership).
On my fourth trip to the DMV, I successfully picked up plates and thought the monumental mess was over. I had exchanged dozens of emails and phone calls with these guys over months and don’t recall ever dealing with people so inept, rude, and unprofessional at a car dealer which is saying a lot because of all the random vehicles I’ve purchased. Anyways about 2 weeks after I got my tags, I get a letter in the mail from the DMV saying the dealership check BOUNCED and my registration was now canceled and if I was pulled over or parked on a public street my 200 would get impounded! My anger and frustration level to this point had been tempered by the fact I had temp tags and could drive it, but after this letter I nearly lost it. It’s a very long and twisted story from this point, but I am convinced through the extremely heated conversations I had the general manager intentionally put a stop payment on the check knowing the ramifications it would have. This specific guy had been called out many times in reviews for some of the shadiest business practices a car dealer could do, and the whole dealership group he worked at had nothing but bad reviews.
The very long story short is I was able to get everything resolved, but it took about half a year and an unbelievable amount of stress and time. My deal was different because it was a used car, so I think yours should be easier to sort out. I personally don’t think a lawyer is going to speed up your process much more than you’ll be able to on your own, and it will definitely be even more money out of your pocket. If I was in your shoes, I’d be tempted to just tell them to rescind the deal, but that will definitely be a huge hassle. I wouldn’t be surprised if the Covid situation has totally screwed up the way the DMVs are operating right now, so maybe give it another week of pressing the issue before you go the lawyer or return route. I’m confident neither party wants to go down that route, so hopefully it gets resolved ASAP.