Gene dosage alterations, such as rare copy-number variants (CNVs), are major drivers of disease risk and whole-body multimorbidity. Their rarity makes deciphering this biological footprint challenging. To overcome this statistical limitation, we developed Functional Burden analysis (FunBurd), aggregating protein-coding CNVs across 172 transcriptomic-derived tissue and cell-type networks. Applying this to 43 complex traits in ~500,000 UK Biobank participants, we captured widespread functional associations missed by single-gene approaches. Mapping this architecture revealed a unifying principle: CNV pleiotropy is fundamentally restricted by evolutionary constraint and overwhelmingly concentrated within brain-specific functions. Crucially, mediation analysis proved these pleiotropic links are driven overwhelmingly (~84%) by direct CNV effects. This architectural constraint simultaneously governs gene dosage responses. Highly conserved brain functions, intolerant to dosage deviation, exhibited predominantly non-monotonic (same-direction) effects, whereas less constrained non-brain traits followed traditional monotonic rules. Within these brain functions, we uncovered profound sex differences: females exhibited a significantly higher proportion of deletion-driven associations for mental health traits. We successfully replicated these associations in an independent, ancestrally diverse cohort of ~500,000 All of Us participants. Furthermore, deletion burden demonstrated positive effect size correlations (up to r=0.78) with rare pLoF SNVs across 87% of traits, confirming that the effects captured by the functional burden test are consistent with an independent class of loss-of-function variants. Our results highlight the key role of genetic constraint and brain-specific mechanisms in shaping CNV-driven pleiotropy and monotonic gene-dosage response, providing a mechanistic basis for the whole-body multimorbidity observed in neurodevelopmental and psychiatric conditions.