Public Water Supply Invasive Species Compliance Act of 2021 This bill addresses transfers between Texas, Arkansas, and Louisiana of water that contains invasive species. The bill exempts certain water transfers between public water supplies in Texas, Arkansas, and Louisiana from prohibitions on illegal trade of plants and wildlife. Specifically, the prohibitions do not apply to covered water transfers containing prohibited species if the species are present in both public water supplies before the transfer, the water is subject to mitigation measures, and the water is transferred directly between the supplies; or the water is transferred in a closed conveyance system, such as a pipe system, and sent directly to treatment facilities where the species will be destroyed. The costs of the mitigation measures must be borne by the entity that sells the water for financial gain. Finally, the bill establishes notification requirements for water transfers.