In my belief, one can work part-time for the sponsoring company. It has to be EAD, because most likely, H-1B was for the full-time position. The company can file for a half-time position H-1B, by the way.
However, it is crucial to understand that Employment-based Green Card is based on a full-time job offer. So, the sponsoring company while employing you in a half-time job (better with the same job description), should furnish a letter with a full-time job offer for the time whenever your Green Card is ready.
Your case will be aggravated with this twist, no doubt, but a good lawyer can argue that in the current job market companies seek to cut the costs, yadda, yadda... A bad lawyer, however, can make it look as a misrepresentation, which you do not want at all.
Then again, a part-time job with the sponsoring employer is better than no job at all.