Gamma-ray beams with large angular momentum are a very valuable tool to study astrophysical phenomena in a laboratory. We investigate generation of well-collimated $gamma$-ray beams with a very large orbital angular momentum using nonlinear Compton scattering of a strong laser pulse of twisted photons at ultra-relativistic electrons. Angular momentum conservation among absorbed laser photons, quantum radiation and electrons are numerically demonstrated in the quantum radiation dominated regime. We point out that the angular momentum of the absorbed laser photons is not solely transferred to the emitted $gamma$-photons, but due to radiation reaction shared between the $gamma$-photons and interacting electrons. The efficiency of the angular momentum transfer is optimized with respect to the laser and electron beam parameters. The accompanying process of electron-positron pair production is furthermore shown to enhance the orbital angular momentum gained by the $gamma$-ray beam.
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