How do I flip the held item texture but not the inventory one?

Can you send an image of the texture? This sounds odd.

You will need to edit the .json display code of the item and replace it with the following:

{
  "parent": "item/handheld",
  "textures": {
    "layer0": "modid:item/item_texture"
  },
  "display": {
      "ground": {
          "rotation": [ 0, 0, 0 ],
          "translation": [ 0, 3, 0],
          "scale":[ 1, 1, 1 ]
      },
      "fixed": {
          "rotation": [ 0, 0, 0 ],
          "translation": [ 0, 0, 0],
          "scale":[ 1, 1, 1 ]
      },
      "thirdperson_righthand": {
          "rotation": [ 90, -90, 130 ],
          "translation": [ 0, 0, 0 ],
          "scale": [ 1, 1, 1 ]
      },
      "thirdperson_lefthand": {
          "rotation": [ 90, 90, -130 ],
          "translation": [ 0, 0, 0 ],
          "scale": [ 1, 1, 1 ]
      },
      "firstperson_righthand": {
          "rotation": [ 90, -90, 130 ],
          "translation": [ 0, 0, 0 ],
          "scale": [ 1, 1, 1 ]
      },
      "firstperson_lefthand": {
          "rotation": [ 90, 90, -130 ],
          "translation": [ 0, 0, 0 ],
          "scale": [ 1, 1, 1 ]
      }
  }
}

You will then need to replace modid with your mod's id, and item_texture with the internal name of your item's texture file.

Lastly, you will need to edit the rotation values, and possibly the middle translation values, of thirdperson_righthand, thirdperson_lefthand, firstperson_righthand, and firstperson_lefthand.

I'm not sure what exact values you will need, but you should be able to experiment with the provided code to figure it out.

Lastly, be sure to save and lock your code.

I know that they correspond to X, Y, Z rotations, but I can't remember exactly where each axis is located in terms of built-in item model rotations. As I mentioned in my previous comment, I would just recommend experimenting, maybe create three copies of the item and have each copy rotated 90 degrees on a different axis to find out what axis does what.