I was in a similar situation a few years ago. I was sent by a new company to LR-35 Recurrent (I had previously been to FSI LR-35 Initial and had 1300 in type) and the new company was not going to type me, though they had promised me "quick upgrade to PIC" based upon my time in the jet and sparkling smile. Normally, I'd have gone along with that, but after I was hired, I saw the exact jet that I was to fly listed for sale in a magazine! Not knowing how long the jet may be around, and fearing that I would never get another chance at a LR-JET type,I asked about paying the extra. $2,000 was the quote by FSI, but I needed the company authorization as it was a 135 checkride, in addition to my type and ATP. The company caved and paid the difference for me, but turned it into a training contract for twelve months. It was worth it, and I was made PIC a month later.
As for the "Rotate" issue, I can see no reason to delay the rotation beyond Vr unless there is a directional control issue which requires that the nose gear be on the runway.
And here's my tip for those new to jets and simulators: If you should find yourself off to one side of the runway after the engine failure, apply rudder aiming for the center of the far end of the runway. That way, you'll be correcting without over-correcting. My problem was that I aggresively corrected back to centerline and then weaved quite a bit once I got there, which complicated the rotation.
Can the original poster clue us in on what he did regarding his type-rating?