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Wave propagation effects such as resonance and interference effects complicate the design of many acoustic devices, particularly when the dimensions of the device are in the order of the operating wavelength. At the same time, these complications also offer an opportunity for numerical optimization schemes to outperform designs achieved using traditional methodology. An example of a device sensitive to resonance and interference effects is the compression driver, the standard sound source for midrange acoustic horns in public address systems. Although ingenious and rather simple design guidelines have been developed, these unfortunately only apply to particular conceptual compression driver layouts. Here, we address a configuration for which no simple rules exist and apply numerical shape optimization for the design task. We employ a level-set geometry description of the crucial part of the compression driver interior. To avoid mesh changes when the level-set function is updated by a gradient-based optimization algorithm, we rely on the cut finite element (CutFEM) technique for the acoustic modeling. A particular modeling challenge here is that viscothermal losses cannot be ignored, due to narrow chambers and slits in the device. Up to quite recently, the modeling of such losses has required computationally expensive solutions of the linearized, compressible Navier–Stokes equations, making the use of shape optimization extremely challenging. Fortunately, a recently developed, accurate, but computationally inexpensive boundary-layer model is applicable in this case. For the first time in the context of a CutFEM/level-set method, the shape calculus needed to compute derivatives for the optimization algorithm is carried out in the fully discrete case, taking into account the discontinuities along the design boundary of the pressure derivatives and the normal field. Applying these techniques, the algorithm was able to successfully design the interior of a compression driver so that the final frequency response very closely matches an ideal response, derived by a lumped circuit model where wave interference effects are not accounted for.