Bipartisan Border Solutions Act of 2021 This bill establishes regional processing centers for conducting immigration-related activities and addresses other related issues. The Department of Homeland Security (DHS) shall establish at least four regional processing centers along the southern border in high-traffic Border Patrol sectors. These centers shall conduct processing activities such as criminal history checks, medical screenings, asylum interviews and credible fear determinations, and short-term detention of individuals. The bill imposes various requirements, such as requiring the centers to (1) allow nongovernmental entities access to detained individuals to provide humanitarian or legal assistance, and (2) provide detained aliens access to legal counsel for certain asylum proceedings. The bill also requires DHS and the Department of Justice (DOJ) to develop pilot programs to improve asylum decisions, credible fear determinations, access to counsel, and case management of aliens awaiting asylum hearings or decisions. DOJ shall, to the greatest extent practicable, prioritize removal cases involving aliens apprehended while entering the United States during an irregular migration influx. Before placing an unaccompanied alien child with an individual, the Department of Health and Human Services must conduct a criminal background check of each adult member of that individual's household. An alien child may not be placed in a household where an individual has been convicted of certain crimes, including offenses involving sex or domestic violence. The bill also makes it a crime to make a materially false statement or knowingly use a false document to obtain custody of an unaccompanied alien child.