Consider this answer as an addendum, not as a direct answer for the color checker usage.
And I am writing it because your posted images are showing some important deficiencies.
To get the right colors you need a bit more than just the color checker. In fact, it is probably the last step, not the only or primary one.
Not only they need to be in a consistent spot, avoiding reflections and unwanted shadows, but consistency across your photo. Be aware of any fall of because of the distance from the light source.
Your lights should have a good enough CRI, Color Rendition Index, so use some good quality lights. Avoid Fluorescent or cheap LEDs. Use a flash if possible.
Define the white balance of the light setup.
I don't want to make a "profile" for future use with this camera...these specific lighting conditions.
A profile is made for a combination of Camera + Lens + Light setup, including ambient light if it is relevant. If you change any of them, you should redefine at least the white balance. (Or make a new profile since you have already a checker)
Here is the first usage of the color checker. Use the gray targets to define the white balance, but personally I prefer using some other methods to do this, mainly taking the picture of my main softbox.
Here it is another major issue on your light setup, they are way underexposed. You will only have a lot of noise when you adjust it.
You need to know your camera, if your camera gives you a reading of X aperture and Y speed, taking the photo of a white background and render it as gray, you need to know how many stops or speed you need to increase to make the white look white without cropping it. Depending on the camera and the look you want, you could use 3 stops brighter than the base reading. The histogram is your BEST friend.
Position of target
It should be in the middle of the framing your product will be. Put it on the pillow. This is closely related to point 1, fall off, ambient light contamination, etc.
Take the photo and
Yeap, you do not use it only for the image you want to correct. You need to apply the adjustments on the sample image, save a profile and then apply the changes to the other photos on the exact conditions the target image was shot.
The first correction is white balance
But I am afraid I do not know the procedure to apply the other color corrections on free software. But you can apply them directly on Ps, without the need for Lightroom.