Would there be any other symptoms if the balance shaft timing was off? Would it only cause vibrations in such a narrow rpm band? The guy who rebuilt my engine mentioned not knowing how to set up the timing and he would have to get someone else to do it.
Vibration in an engine with balance shafts, check the shaft timing FIRST. That's 10:1 gonna be your problem.
I can't tell you about this model or engine but as a generality these shafts have little counterweights and spin at a multiple of engine RPM. They're designed to cancel out harmonics of the (unavoidable) imbalance in a normal ICE. However this only works if the counterweights are correctly synchronized so they
cancel the built-in vibration. They can easily be installed so they
add to the vibration and then you've got exactly what you've (probably) got.
Generally the balance shafts are driven from the timing belt so fixing wrong synch is done by accessing the belt, slacking it off, following the directions to put the shaft(s) in the right place, and tightening the belt again.
You may be able to check correct/incorrect timing without digging very deeply. On the Mitsubishi 4G64 2.4L engines there's a plug on the left side of the block, you take that out, set the engine to TDC and put screwdriver in the hole you took the plug from If it goes in 60 mm you're okay, if not the shaft is wrong. (Or the reverse -- I haven't done this lately)
The problem on that engine is that that shaft is driven by the oil pump sprocket but goes at half the OP sprocket RPM. So when the sprocket is correctly aligned (as you put the belt on) the balance shaft can either be right or 180 out and you can't see the shaft -- it's in the crankcase. You must try it, stick in your screwdriver, and if wrong, turn the OP sprocket another full turn.
I gave up trying to explain this to mechanics. Those who don't already understand it and get it right, never will.
Anyway, search something like 'vibe/2.4l 2az fe check balance shaft timing' and go from there. This is a
must fix problem.
Yes, a balance shaft out of time will kill engine mounts. Plus no telling whatall else if allow to persist.
EDIT: Okay, I've followed my own advice and done the search. Looks like those balance shafts are driven by gears. And the drive is in the crankcase with the shafts. That's a much tougher alignment job than my Mitsubishis. I would take that car to a RAV-4 dealer, show him the symptom, and ask if he thinks it would be worth checking the balance shafts, and what would be the price. Yes, don't bother family about it.