Is there a way to suppress *all* Dynare output in Matlab?

Yes, you should be looping over stoch_simul using the construction outlined in
[Loop over parameters)
Whenever resol (or stoch_simul) is called, the steady state is recomputed.
You can also work from the replication codes to Born/Pfeifer (2014): “Risk Matters: A comment” at aeaweb.org/articles.php?doi=10.1257/aer.104.12.4231. Essentially, instead of solving the model and simulating it within the objective function to compute the distance between simulated and empirical moments, you retrieve the model IRFs and then compute the distance.

I plan on providing an example of this soon.