The CMP is simply a program. The law that established the DCM- which later became the CMP- AFAIK do not trump state law in any way.
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...federal law trumps state law .
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It is not that simple. Up until the
D.C. v. Heller decision, it was a basic assumption of most federal-level firearms law that the states had an externally unchecked* power to restrict the possession, ownership, and carrying of firearms within their own borders. The Commerce Clause gave the Feds the power to pass minimum standards and taxes that affected interstate commerce in arms, but the states could enact whatever additional standards they saw fit to enforce within their own borders, and those regulations were not the Feds' business. Most federal firearms law is written with this in mind.
IOW most federal firearms law adds restrictions, but is carefully written to avoid overturning or preempting more stringent state restrictions.
*Many states have a counterpart to the 2A enshrined in the state constitution, but my point is that the checks on the states' powers to regulate arms came from within, rather than from without.