  The next generation of weak lensing surveys will measure the matter
distribution of the local Universe with unprecedented precision, allowing the
resolution of non-Gaussian features of the convergence field. This encourages
the use of higher-order mass-map statistics for cosmological parameter
inference. We extend the forward-modelling based methodology introduced in a
previous forecast paper to match these new requirements. We provide multiple
forecasts for the wCDM parameter constraints that can be expected from stage 3
and 4 weak lensing surveys. We consider different survey setups, summary
statistics and mass map filters including wavelets. We take into account the
shear bias, photometric redshift uncertainties and intrinsic alignment. The
impact of baryons is investigated and the necessary scale cuts are applied. We
compare the angular power spectrum analysis to peak and minima counts as well
as Minkowski functionals of the mass maps. We find a preference for Starlet
over Gaussian filters. Our results suggest that using a survey setup with 10
instead of 5 tomographic redshift bins is beneficial. Adding cross-tomographic
information improves the constraints on cosmology and especially on galaxy
intrinsic alignment for all statistics. In terms of constraining power, we find
the angular power spectrum and the peak counts to be equally matched for stage
4 surveys, followed by minima counts and the Minkowski functionals. Combining
different summary statistics significantly improves the constraints and
compensates the stringent scale cuts. We identify the most `cost-effective'
combination to be the angular power spectrum, peak counts and Minkowski
functionals following Starlet filtering.
