To obtain an accurate cosmological inference from upcoming weak lensing surveys such as the one conducted by Euclid, the shear measurement requires calibration using galaxy image simulations. As it typically requires millions of simulated galaxy images and consequently a substantial computational effort, seeking methods to speed the calibration up is valuable. We study the efficiency of different noise cancellation methods that aim at reducing the simulation volume required to reach a given precision in the shear measurement. The more efficient a method is, the faster we can estimate the relevant biases up to a required precision level. Explicitly, we compared fit methods with different noise cancellations and a method based on responses. We used GalSim to simulate galaxies both on a grid and at random positions in larger scenes. Placing the galaxies at random positions requires their detection, which we performed with SExtractor. On the grid, we neglected the detection step and, therefore, the potential detection bias arising from it. The shear of the simulated images was measured with the fast moment-based method KSB, for which we note deviations from purely linear shear measurement biases. For the estimation of uncertainties, we used bootstrapping as an empirical method. We extended the response-based approach to work on a wider range of shears and provide accurate estimates of selection biases. We find that each method we studied on top of shape noise cancellation can further increase the efficiency of calibration simulations. The improvement depends on the considered shear amplitude range and the type of simulations (grid-based or random positions). The response method on a grid for small shears provides the biggest improvement. Here the runtime for the estimation of multiplicative biases can be lowered by a factor of 145 compared to the benchmark simulations without any cancellation. In the more realistic case of randomly positioned galaxies, we still find an improvement factor of 70 for small shears using the response method. Alternatively, the runtime can be lowered by a factor of 7 already using pixel noise cancellation on top of shape noise cancellation. Furthermore, we demonstrate that the efficiency of shape noise cancellation can be enhanced in the presence of blending if entire scenes are rotated instead of individual galaxies.