Hi daddydaycare. Welcome to Mrs. Clean.
It sounds like this might be a tough stain. You do not mention the color of the makeup stain on the carpet, but since it is kids makeup it is likely to be strong and bright- Most likely a very strong dye in an oil base. This is a bad combo with carpets.
I have some tips to help you out on this. You may need to call a professional if these do not help. Please remember to be very gentle when working the stain so you do not fray the carpet fibers. Also test any products first in an inconspicuous place. I'd also recommend using cold water when extracting the stain to avoid heat setting any dye in the carpet fiber.
Here are solutions for removing oil based dye's (lipstick, for example) using either rubbing alcohol, hairspray, or a commercial solvent. I'd try using a solvent first to dissolve the oily stain, before using a detergent. Detergents are good removing oil and grease, but leave a residue in the carpet which attracts further staining.
How to remove lipstick stains.
After you have removed the grease part of the stain, if any of the dye remains, you can try this method for lightening any stain.
Good luck