What is the CPSIA?

Yes, with a couple of caveats.

Here is a list of materials that you do not have to send to a laboratory in order to prove that they are lead-free.

What information has to be put on the label or alternatively, somewhere on the product and/or its packaging?

the name of the manufacturer or private labeler
the location of production (city, state/province, and country)
the date of production OR a mark, such as a batch number or cohort number, that you can use to look up the date [...]

If your product has multiple pieces, you do not need to label each piece.

If you already keep records of your materials and sources and you can tell where each material came from, you do not need to start using a system of batch numbers.

Products are exempt from labeling when it is not “practicable” to label them.