This institution doesn't allow us to do this or any alternative ways to do this.
Also, can I get J1 waiver and move to H1B if the position is funded by NIH?
-Yes if they hire you. Unlkess the job require US citizen or GC holder specifically, NIH sponsor the H1 visa or J1 visa.
Does my child need to go back to China if I get J1 and he needs to change to J2?
-He can change his status to J2 within US. You also can change your F1 to J1 within US. But my suggestion is that you may just wait for the H1 approval to work if the employer allows you to wait. It may take about 3 months. There are not many H1 applicants due to the bad economy, so the H1 petition may be approved faster than we expect.