{tmap} places labels in centroids, and there is not much you can do with it; this is a big issue with Prague / Středočeský kraj. Auto.placement is random, and thus not easily reproducible (set.seed() may help). xmod and ymod are applied to all labels equally, so will not help with overlap.
If you are not dead set on using {tmap} you may consider plotting via {ggplot2} and use {ggrepel} - this is what I would do.
Also, consider transforming your coordinate reference system to WGS84, so that north is up; the Křovák projection looks kind of strange 