If someone complains about a “diesel” sound, my first suspect would be some kind of exhaust problem. The exhaust sound is not being properly muffled. This sound can vary, from like firecrackers going off, to a put-put-put sound. It depends on where it is sourced, and how open the sound is to the environment. So I’d start probably by suspecting a bad muffler, next, a cracked exhaust pipe or cat, next, a cracked exhaust manifold, or finally, a problem with an exhaust valve not working properly.
It sounds like your shop did some initital tests, presumably ruled out most of the items above, and decided it to be a valve problem. At the point they said they couldn’t locate the proper 2nd cam, to me, that would be a red flag. It shouldn’t be difficult to find a used one using the junkyard computer network.
I guess if this were my car, the first thing I’d do is tow it away from the dealer shop asap. And tel the tow driver to take it to an inde mechanic for a second opinion. A dealer shop is good at warranty repairs and diagnosing drivability problems in out of warranty vehicles, but isn’t the best choice for actually doing major engine repairs like this. You’ll get better results at an inde mechanic. Ask friends, co-workers etc for a recommendation.
And if you are looking at $8,000 for a used/rebuilt engine, unless you are confident that the rebuilder has the experience, tooling, and testing to insure the rebuild job delivers a “new” engine, you may well decide based on cost considerations it makes more sense to send this vehicle to the recyler, and buy a new one.