Qualitative Color Palette - photocopy safe (grayscale ready)

As ColorBrewer suggests only the 5 class spectral color scheme is "safe" for photocopier (i.e. grey scale) use. The easiest solution is to use some other way of distinguishing the classes such as hatching.

In QGIS it is a simple matter to choose 5 different hatches in a classification by changing the fill brush.

Or by using a Point Pattern fill with different distances:

Have you had a look at the viridis family? Easiest way to add them in QGIS is via this link - http://rocksandwater.net/blog/2016/07/qgis_perceptually_uniform_colorramps/

Colorbrewer 2.0 basically tells us that there is no "photocopy safe" color scheme with 5 colors, so here is the 4 color solution:

Of course, it does not count "white" as a potential 5th color so this solution might still be sufficient for your application.