No, you should use neither a F.4852 nor a F.8919 for this situation.

F.4852 is used to report wages your client's employer failed to report the correct amount of wages or issue a W-2.  It is not for payment of any shortfall in FICA.  There is also no mechanism for the IRS to retain voluntary payment of FICA.  Any excess credit resulting from such voluntary payment will just be refunded to your client.

F.8919 should not be used.  It is only for folks who believe that they were misclassified as an independent contractor when they are, in fact, employees.  Submitting this form will most likely result in rejection or additional complications as the IRS is very mechanical in how forms are processed and by which unit.

Only the employer can correct the error.  By refusing to correct the withholding and reporting, the employer is in breach of its fiduciary duty to withhold and report as well as to pay over FUTA.  Any unpaid FICA from your client could be a liability of the employer according to the IRC.  The bottom line is that §3121(b)(19) applies only if your client was a nonresident alien and a J visa holder, not if she was a resident alien (under §7701(b) by acquiring a green card, failing to be an exempt individual and meeting substantial presence test, etc.).  In case these talking points still do not convince the employer, her only recourse would be to contact the IRS to file a complaint.