Can I add third party lens profiles to Nikon cameras or software?

If you're willing to do an intermediate step using 16-bit TIFF files, the best way to get where you want is to:

• Shoot .NEF, then use Nikon's software to process your color, contrast, curves, HSL, etc. before outputting from the Nikon application to 16-bit TIFF. The color and other processing you do in the Nikon software will be "baked in" to the TIFF files. It's just that with a 16-bit raster image you'll have more latitude for additional adjustments in Photoshop or whatever other graphics processing software you choose to use than with an 8-bit JPEG.
• Import your TIFFs into Photoshop (or Gimp, or Darktable, etc.) and apply correction for lens distortion, CA, etc., there.
• If you have an image that you wish to process using multiple layers, you might need to export multiple TIFFs from Nikon's Capture with the brightness, contrast, colors, etc. adjusted for what you desire in each layer. Then combine them using each TIFF for each layer of the image in Photoshop or other graphics processing software.
• Another approach to such high dynamic range scenes would be to adjust everything to your liking in Nikon's Capture, then expand the contrast to include all shades in the raw file before exporting. Sure, it will look "flat", but all of your color adjustments will still be in there and should look very similar when you raise the black point and lower the white point to around the same levels in Photoshop.

There's a lot to keep in mind about raw vs. TIFF:

The 12, 14, or 16-bits used to record raw data and the 16-bits (per color channel) used to record a demosaiced and gamma corrected TIFF or PSD are not used to represent the same exact thing in the same exact way, even though raw files often use containers that conform to the larger TIFF standard.

Raw files have one single, monochromatic brightness value for each photosite (a/k/a sensel or pixel well). Each sensel has a color filter over it that is one of three colors that are often referred to as "red", "green", and "blue. But the colors used in Bayer masks are NOT the same colors as the 'Red', 'Greeen', and 'Blue' colors emitted by our RGB display systems. To get any real color information out of the information contained in a raw file, the monochrome luminance values measured by each sensel are compared to the monochrome luminance values record by surrounding sensels filtered with each of the colors used in the Bayer mask and color information is interpolated from the results. This works as well as it does because the human vision system works in a remarkably similar manner. The wavelengths that the three types of cones in our eyes are most sensitive to are not 'Red', 'Green', and 'Blue' either!¹

What we usually mean when we say "TIFF files" are raster images that have a 16-bit number for each of the red, green, and blue channels for each pixel in the image. That's why 16-bit TIFF files are so much larger than 14-bit raw files, even though they actually contain less total information.

For more, please see:

Why can software correct white balance more accurately for RAW files than it can with JPEGs?

RAW to TIFF or PSD 16bit loses color depth

Why are Red, Green, and Blue the primary colors of light?

¹ The colors used in Bayer masks are not the same colors emitted by our RGB screens. "Red" is not even close. The "red" cones in our retinas are most sensitive to about 564nm (a lime-green color between green and yellow), the "red" filters in most Bayer masks are most transmissive to about 590-600nm (orange-yellow), while our RGB systems emit 640-650nm for 'Red'.