This answer has some good news and some bad ones.
The color checker uses a 24 color Macbeth chart. So you can use another brand if you want https://www.google.com.mx/search?q=macbeth+chart and use the internal Adobe software for color profiling. The color checker has its own software but I am not sure if works exactly the same. (I will try to detail this later)
A Pantone book could be used as "a reference". But there is no exact equivalent to the values of the chart, and the "uncoated" still has a glossy finish due the nature of the pigments and materials used. (I'll try to post some images later)
If you do not have an "exact color" the software used will tweak the image thinking the target is correct but the light is wrong... which will give you unpredicted results.
Besides the fact that you would need to "chop" your Pantone guide to make a fake Macbeth chart... That is not a good idea because the Pantone book could be more expensive than some brands of the color chart.
Would this be helpful in color correcting? Is there some method to matching between CMYK and RGB formats in this case?
Ouch. Here is a totally different topic than calibrating your camera/light setup.
An RGB-CMYK "match" is, in reality, a conversion that depends on the color profile, the type of inks, the calibration of the device, the paper used.
This also uses a series of patches but it needs to be measured by a colorimeter.
This renders a totally new set of problems.
I noticed the X-Rite states it works across lighting conditions
It does not "work" across lighting conditions, it is to "profile" those lighting conditions. You need to do a test shot whenever you change your light sources, regardless you use the same camera. Also, you can profile different cameras using the same light setup for example.
This is a camera/light profiling methodology.
For a fast "color correction" do a white balance shot. Period.
How to make a good white balance test shot?
If you are using one diffuse light you can shot it as a target.
You can use an Ink Jet paper marked with the highest "whiteness" you can find. Use a new one. Old paper tend to turn yellow.
Underexpose this shot. In reality, the "white balance" should be called the "gray balance".