use of ~

Is the use of "*bʰér-ti-s ~ *bʰr̥-téy-s" in Reconstruction:Proto-Indo-European/bʰer- a sort of enunciation of NOMINATIVE ~ GENITIVE for Reconstruction:Proto-Indo-European/bʰértis.

Yes. The main purpose is to show the alternation in the stem as a result of ablaut, since many times the descendants preserve the genitive stem rather than the nominative stem.