The most effective training method I have employed over the years with students who have issues over controlling the simulator on V1 cuts was a simple rhyme Squeeze, Freeze, 12 Degrees. This pitch initial attitude is applicable to Boeing 737 600 through 800 series.
Squeeze - the rudder to parallel and not necessarily regain centreline. If recognition is prompt enough the drift from centreline should be minimal and trying to correct back to centreline in the short timeframe between V1 and VR IMHO causes more issues of over controlling than it solves.
Freeze - Once the yaw is controlled and the aircraft is paralleling the centreline, freeze that rudder input. At rotation some aileron will be required to maintain wings level during initial climb out and at this time the aircraft may briefly cross-controlled. With the rudder input frozen it is much easier to finely adjust the aileron input to maintain wings level. This input should never exceed 5 degrees.
12 Degrees - Rotate smoothly and slowly (Boeing recommends 1½ - 2 degrees per second as opposed to the normal 3 degrees per second) to an initial target pitch of 12 degrees (10 degrees on the B737-200) using ADI/PFD raw data initially and not FD cues. Once the gear is retracted, with a positive rate of climb, and pitch is stabilized at at an attitude that results in V2 (don't chase the airspeed, allow it to stabilize and then make corrections) FD commands can be followed and fine adjustments can be made to rudder and aileron inputs to accomplish co-ordinated flight.
From my experience most over controlling occurs as a result of far too aggressive rudder inputs both on the runway and in the air and if treated in the same fashion as PIO on short final (freeze all inputs momentarily then make adjustments in a co-ordinated manner) V1 cuts in the simulator can be transformed from a very stressful event to a non event in fairly short order.
Once the basics have been mastered, then other complications such as weights, density altitude and varying CofG can be introduced.