My '48 truck did not have a data plate or an original engine (they used to go by the numbers on the engine), but I did have a title so it was easy to transfer and register. The result is I have a truck, I have a title, but there isn't any stamped number on the truck that ties it to the title. Here at the Arizona DMV they didn't even want to look to see if the title matched the vehicle's manufactures number, so it was no problem.
Motor vehicles didn't have standardized VINs until after 1954 For my GMC, the engine had the manufacture's number on it, and that's long gone. When I bought the truck, I transferred the title to my name, no problem - they didn't even look at the truck (thank goodness since most of it was in boxes)
The only problem I had when I registered it after sitting several years during the restoration was that I had to provide an affidavit that it hadn't been driven and pay a small fine. They didn't want to look at the truck, just the paperwork.
I've built or bought cars that I had to get a 'bonded' title (I think this is what you're referring to as a 'lost title') In Arizona, this requires that you post a 'surety bond' that you can purchase from most independent insurance brokers (my last one cost $150). More important than the surety bond is the DMV's requirement of a "Level 2" inspection at one of their local offices. They are going to want to look at the VIN for evidence of tampering - DO NOT, UNDER ANY CIRCUMSTANCES SHOW UP TO A DMV INSPECTION WITH A VIN PLATE THAT DOES NOT LOOK ORIGINAL TO THE TRUCK. The bureaucrats at the DMV are looking for evidence of VIN fraud, and they take their job very seriously.
I'd transfer the title I already had. You can go back and change the data plate later to match if you want. Remember this is coming from someone next door in AZ, but DMVs operate more-or-less the same.
If you want to hear the back story of my DMV adventures, keep reading..............
A few years ago I bought a 1974 Triumph TR6 without a title. I completely restored the car, including putting over 400 hours in paint and body work, but I didn't bother to get a title until the car was roadworthy again. I'd built several off-road vehicles from scratch before and used the title bonding process to get them titled, so I figured it'd be simple. (all I had to do was jump through some hoops to prove the major components weren't stolen - but that's another story)
The car had an aluminum VIN plate in the engine bay. During the paint job, I removed it to sandblast the engine bay. I replaced it with a new polished VIN plate and had the number engraved on it, then re-riveted it to the firewall. Looked tits.
Having completed the car, I drove it down to the DMV with my application and surety bond in hand, ready for the 'level 2' inspection. The inspector said "show me the VIN" so I popped the hood to show it to him. He looked at the VIN and said "wait here for a minute, I'll be right back"
He returned a couple minutes later with a claw hammer and an old wood chisel and proceed to try and remove the VIN plate rivets. He'd gotten one lick in with the claw hammer before I managed to talk him into stopping and call his supervisor to see if he'd intervene. The inspector said his intention was to confiscate the 'fake' VIN plate and impound my newly restored car for VIN fraud - he was pretty excited to have busted a major crime syndicate that dared try get an obscure British sports car titled. IIRC correctly the supervisor said if I could produce the old VIN tag and it matched a number on the chassis somewhere, he'd let me go ahead with the title application. I managed to take the car back home that afternoon with a bonded title, but I could have lost my car. Instead of the factory VIN, I had to have an 'assigned VIN' that the DMV applied to my chassis on a metal sticker. (Sold the car to a guy that exported it to Austria and didn't seem to care if the state and manufacture's VINs matched)