I am foreign graduate and have done two years of preliminary GS and 1
year of research on an H1 visa. Whenever I apply, the program does not
entertain my application because I do not have a J1. Even when I
explain to them that my current status does not preclude me from
getting a J1 for their start date, they do not listen.

Even if I had a J1 it would have been specific for my institution. I
would have to get a new J1 anyway. That basically means going through
the visa process all over again to get a new visa.

In short: current visa status (H1,B1, B2, F1) has hardly anything to
do with the status at the beginning of training. Foreign graduate
cannot get a J1 for the new position unless the program gives them a
contract and the ECFMG processes the application after that. A current
J1 is immaterial since it is institution specific.

The institutional policies based on lack of understanding of the visa
issues that have only served to keep out reasonably qualified
candidates.

What do I do? I need suggestion and help.

Did you try writing personal emails to all those programs that ECFMG
will sponsor your J1 visa for the program  , which will be granted
AFTER the program agrees to endorse u for the program ..