Viruses are under relentless selective pressure from host immune defenses. To study how poxviruses adapt to innate immune detection pathways, we performed serial infections of vaccinia virus in primary human cells. Independent courses of experimental evolution with a recombinant strain lacking E3L revealed several high frequency point mutations in conserved poxvirus genes, suggesting important roles for essential poxvirus proteins in innate immune subversion. Two distinct mutations were identified in the viral RNA polymerase gene A24R, which seem to act through different mechanisms to increase virus replication. Specifically, a Leu18Phe substitution in A24R conferred fitness tradeoffs, including increased activation of the antiviral factor Protein kinase R (PKR). Intriguingly, this A24R variant underwent a drastic selective sweep during passaging, despite enhanced PKR activity. We show that the sweep of this variant can be accelerated by the presence of copy number variation (CNV) at the K3L locus, which with multiple copies strongly reduces PKR activation. Therefore, adaptive cases of CNV can also facilitate the accumulation of point mutations separate from the expanded locus. This study reveals how rapid bouts of gene copy number amplification during accrual of distant point mutations can potently facilitate poxvirus adaptation to host defenses.