Unfortunately, it isn't an issue of fighting it. It either is a cap-exempt institution or it is not a cap-exempt institution. The law in this regard is found under INA § 214(g)(5) - (11), but primarily, you will look here:
(5) 2/ The numerical limitations contained in paragraph (1)(A) shall not apply to any nonimmigrant alien issued a visa or otherwise provided status under section 101(a)(15)(H)(i)(b) who 2d/ --
(A) 2e/ is employed (or has received an offer of employment) at an institution of higher education (as defined in section 101(a) of the Higher Education Act of 1965 (20 U.S.C. 1001(a))), or a related or affiliated nonprofit entity; 2e/
(B) 2e/ is employed (or has received an offer of employment) at a nonprofit research organization or a governmental research organization; or 2e/
And here is the link with the definition of an institution of higher learning:
(a) Institution of higher education
For purposes of this chapter, other than subchapter IV, the term “institution of higher education” means an educational institution in any State that—
(1) admits as regular students only persons having a certificate of graduation from a school providing secondary education, or the recognized equivalent of such a certificate, or persons who meet the requirements of section 1091 (d) of this title;
(2) is legally authorized within such State to provide a program of education beyond secondary education;
(3) provides an educational program for which the institution awards a bachelor’s degree or provides not less than a 2-year program that is acceptable for full credit toward such a degree, or awards a degree that is acceptable for admission to a graduate or professional degree program, subject to review and approval by the Secretary;
(4) is a public or other nonprofit institution; and
(5) is accredited by a nationally recognized accrediting agency or association, or if not so accredited, is an institution that has been granted preaccreditation status by such an agency or association that has been recognized by the Secretary for the granting of preaccreditation status, and the Secretary has determined that there is satisfactory assurance that the institution will meet the accreditation standards of such an agency or association within a reasonable time.
So the only way you can “fight” it is if you can prove that your institution fits into (a). If it does not, then you would be wasting your time and money. But if you can make a viable argument that it does, then go for it. If not, then you should focus on getting an OPT extension if you can. I know it is not the news that you want to hear and very inconvenient, but the law in this area is complicated and primarily geared to protecting the U.S. job market. I am truly sorry. I wish I had better news for you.
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