Wait, that is illegal. Permanent residents cannot be discriminated against based on what documents they present. You can report this to the
Office of Special Counsel for Immigration-Related Unfair Employment Practices.
An expired green card plus an extension letter is a valid List C document, which combined with a List B document (an ID like driver's license or passport), must be accepted. See
here ("A form I-797 issued to a conditional resident may be an acceptable List C(8) document in combination with his or her expired Form I-551 (“green card”).") and
this handbook (page 16 in the PDF/page 12 on paper, "1. If an employee presents an expired Permanent Resident Card along with a Form I-797, Notice ofAction, that indicates that the card is valid for an additional year, this combination is acceptable List C evidence of employment authorization for one year as indicated on Form I-797 .").
Or you can go into a USCIS office and get an I-551 stamp now based on the immediate need due to employer refusal. (Foreign passport containing temporary I-551 stamp is List A document; though if the employer is not familiar with documents then they may not know what this is either.)
Another way to solve this problem is that she can just use her unrestricted Social Security card, which is a List C document (permanent residents are entitled to a Social Security card without the restriction text; if she has an old Social Security card with restriction text, she can get it updated ASAP), plus a List B document (an ID like driver's license or passport), and doesn't need to present a green card at all.
