YES, you are required to have an inspection by a DC Government inspector. While you may have had a private home inspection when purchasing your property, DC Government requires its own visual inspection by a DC inspector when obtaining a Basic Business License for rental.
Additionally, if you are applying for a Certificate of Occupancy (CO), DC may require a separate inspection as part of that process, in addition to the license inspection. When a CO inspection is required, it must also be conducted by a DC Government inspector.