Thank you to the Special K and Nier Modding Discord servers for helping me out.
Extra thanks to DevolasRevenge for making the Goth Kaine mod in the first place. Without the mod installation guide there I would never have figured this out

Installation guide (heavily based on the aforementioned Goth Kaine mod's guide):
1. Download and local install Special K. Here is a tutorial. Downloading the latest version should work fine.
For Step 5 in the tutorial, Nier Replicant uses a 64-bit executable and the DirectX 11 API. 
For Step 7, rename the DLL file to d3d11.dll. dxgi.dll will not work

2. Download the mod file you chose. Pick one texture folder within the mod file.
Find the file location of your game (the same place you put d3d11.dll earlier). Then, starting from there, place your chosen texture folder in SK_Res\inject\textures. If you are missing folders, create them

3. Start the game, load up a save, and then press CTRL-SHIFT-BACKSPACE to open the SpecialK menu. Click on "Render Mod Tools", then click on "Reload All Injected Textures"

• "Reload All Injected Textures" should be beside "Clear Debug". It should show up, but if you can't see it, go to the bottom right corner of the window and drag the little triangle. That should allow you to resize the window so you can reach the button.
  

The YorHa outfits should have changed color. If not:

• You may have put the texture folder in the wrong place. In the Special K menu go to File -> Browse Texture Assets -> Injectable Textures. This will open up the location the texture folder needs to be in for it to work. It will still work if it's in a subfolder. Restart the game afterwards if you're moving the file over while it's still running.
  

• From what I've personally seen, you don't need to downgrade your game for the texture injection to work, but it might be worth a shot, so here's the exe file for an older build. This link is directly taken from the Goth Kaine mod.
  

• Feel free to ask me questions! I probably forgot to add something
  

Fix for Cutscene Bro Nier and Alternate Kainé models: 
When the aforementioned character is on screen, open the Special K menu and click Render Mod Tools. Find the hexadecimal name of the texture in the list. You may have to click "All Textures" and then "Refresh Textures" for it to show up.
Make the window large enough for you to see a picture of the texture. Then, if the "Reload Texture" button (NOT the "Reload All Injected Textures" button) is available for that texture, click it, and it should work, at least for a while.
If the button is not available/you only see a "Dump Texture to Disk" button, then the fix is not available this time. It's a bit of a toss-up, unfortunately. If the texture disappears entirely from models that already had it successfully injected,  restarting the game will fix it.