Yes - a Rolex that has been modified. If he is happy with how the new Timex movement works, and he doesn't try to pass it off as anything but what it really is, it doesn't matter. In fact, he may even argue that it is better with that Timex movement, because it wont cost him a small fortune to get it fixed if it ever breaks. Of course, it may be worthless and blasphemous to a diehard Rolex afficianado since it is no longer 100% original. But it still has worth to the owner, and to many other folks out there.
The Plymouth is still a Plymouth, regardless of what is under the hood. Cars are primarily known for the bodies, not the engines within.
Jensen Interceptors of the late 60s & early 70s were all factory equipped with either 383 or 440 Mopar engines - does that make them Chrysler cars? No, they're still Jensens.
Look, preferences are like belly buttons (remember those?) - everybody has one. You are not crazy about MoPar cars with non-MoPar engines. I get that, and it is perfectly understandable.
But to imply that the owner of that custom Plymouth is somehow embarrassed by his Chevy drivetrain? Nope, I don't see it.
That doesn't look like a car that has anything to be embarrassed about.
Disclaimer: My 47 Plymouth Special Deluxe Club Coupe has a Corvette engine and a Turbo 350, so my view may be somewhat biased!
Best Wishes to you and to the Spirit of Lively Discussion of All Topics Automotive!