Better Military Housing Act of 2019 This bill requires the Department of Defense (DOD) to develop and implement a plan to address health, safety, and quality issues at privatized military housing. DOD shall provide a tenant entering into a tenancy agreement for such a housing unit information on tenant rights; expectations regarding reporting by tenants and landlords of maintenance, health, or safety issues relating to the unit; a comprehensive accounting of the rights and responsibilities relating to maintenance; and a comprehensive maintenance, repair, and remediation history of the unit. At least annually, the DOD Inspector General shall inspect the oversight of privatized military housing at no fewer than 15 randomly selected installations. Additionally, DOD shall study ways to improve privatized military housing and the oversight and execution of contracts in order to improve housing conditions for Armed Forces members and their families.