The balance shaft, two of them actually, are solid pieces of steel that are driven by a chain from the crank sprocket, so I am wondering what the dealer thinks is wrong with them. Maybe the bearings? Maybe the chain broke? If the chain broke, I would think that it would make a lot of noise and possibly tear up teh oil pump.
Have you had this car since new?
If it is a recent purchase, is there any evidence that the engine may have had some “performance” improvements done? I.e. large exhaust, cold air intake (CAI). Often after the CAI and fart can exhaust, the next modification on this engine is to remove the balance shafts, that frees up about 11hp.
Is this a manual transmission? Another mod that will make it run rough at this RPM is an aluminum flywheel. The stock flywheel is a dual mass flywheel and it is possible that it is worn out too but I’m not sure how that would affect the vibration, it does make using the clutch much more difficult. NA with an automatic transmission.
If it is the balance shaft, you do not need a new engine. Your dealer may not want to do the disassembly on this engine, but it can be done. It can even be done with the engine in place but its no fun. In a shop, I would pull the engine out first. There is a lot of disassembly involved. The balance shafts are located in the upper oil pan, so that means the valve covers have to come off, then the lower oil pan, then the upper oil pan and finally the timing chain cover/oil pump has to come off. The timing chain has to be removed and then the balance shaft assembly can come out for replacement.
This is at least an all day job for one mechanic, maybe more so the labor cost is going to be high. For that reason, I would like to know why he suspects the balance shafts in the first place.