The lack of part throttle is not your problem. You say it only holds the gear change until 2500 RPM max into 2nd. If that's correct, there's another issue. Good luck.
At Wide open throttle in drive, it will hold till 2,500 rpm max to shift into 2nd, then in 2nd gear at wot it will hold till 3,000 rpm then shift to 3rd.
