My wife's employer said that she was exempt from FICA taxes and neither didn't withhold Social Security and Medicare from her wages nor reflected those in W2. (while total wages and Federal income tax withholdings are reflected).

As we are definitely not exempt from the FICA taxes for 2019 (as we pass the substantial presence test being the third year on J1 visa AND we received a green card at the end of 2019), we need to pay that additional Social Security/Medicare on that part of her wages.

When I just manually add info to boxes 3 and 5 on her W2, Turbotax won't add Social Security/Medicare tax liability to the totals.

How should I reflect the liability and pay the tax, then?

Adding it as self-employment income is incorrect because those were wages.

The employer refuses to correct W-2