NSF font research.gov

The LaTeX Times fonts (mathptmx, or better, the newer txmath and txtext fonts) are clones of Times New Roman. My experience (as your earlier experience is) is that this is an acceptable font. I have not submitted an NSF grant through Grants.gov recently enough to know if you will be rejected.

If you will be rejected using the Times clones (because the system is simply trained to look at the font names in the document), then the safest solution is to use LuaLaTeX or XeLaTeX and use Times New Roman as the font (assuming you have it on your system.) The change from your existing document isn't very substantial. Instead of

usepackage[T1]{fontenc}
usepackage{mathptmx}

you can use

usepackage{fontspec}
setmainfont{Times New Roman}
usepackage{newtxmath}

The newtxmath math fonts are a better version of mathptmx. Since the system allows the use of other fonts for math or special symbols/languages the presence of Times New Roman as the main text font should suffice to allow it to pass.

Alternatively, you could use unicode-math and the STIX math font:

usepackage{amsmath}
usepackage{fontspec}
usepackage{unicode-math}
setmathfont[]{Stix Two Math}
setmathfont[range=mathit/{latin,Latin,greek,Greek}]{xits-math}
setmainfont{Times New Roman}