I think the confusion is the word sync which usually means flash. The camera has no mirror lock up unfortunately.
One thought to further diagnose it would be to (with lens off camera) use B to have the camera lift the mirror up and stay while you hold it open. Then with either tape or a toothpick, keep it so it does not return when you release the shutter.
If it stays up, put lens on and take a photo.
Then go back to B and remove mirror prop or tape.
I am wondering if the mirror foam would be sticky enough to cause it to hesitate at faster speeds. Which, it the picture was ok in the new test, might be as simple as replacing the foam yourself.
If its the shutter, then you can anticipate the cost and decide what to do. Keep in mind any replacement might develop the same problem after you own it so fixing it might be an option to consider.
Anyway, good luck to you.