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We report on an approach to produce quasi continuous-wave (cw) electron beams with an average beam current of milliamperes and a mean beam energy of a few MeV in a pulsed RF injector. Potential applications are in the planned laboratory astrophysics programs at DESY. The beam generation is based on field emission from a specially designed metallic field emitter. A quasi cw beam profile is formed over subsequent RF cycles at the resonance frequency of the gun cavity. This is realized by debunching in a cut disk structure accelerating cavity (booster) downstream of the gun. The peak and average beam currents can be tuned in beam dynamics simulations by adjusting operation conditions of the booster cavity. Optimization of the transverse beam size at specific positions (e.g., entrance of the plasma experiment) is performed by applying magnetic focusing fields provided by solenoids along the beam line. In this paper, the design of a microtip field emitter is introduced and characterized in electromagnetic field simulations in the gun cavity. A series of particle tracking simulations are conducted for multi-parametric optimization of the parameters of the produced quasi cw electron beams. The obtained results will be presented and discussed. In addition, measurements of the parasitic field emission (PFE) current (dark current) in the PITZ gun will be exemplarily shown to distinguish its order of magnitude from the produced beam current by the designed field emitter.