Is it permissible to refrain from performing the obligatory Hajj with one's husband due to his continuous emotional abuse throughout the years of marriage?

It is not permissible for a woman to travel for Hajj or anything else except with a mahram and the permission of her husband. The husband is not obliged to perform Hajj with his wife or to spend on her unless there is a condition. A mahram is every trustworthy, sane, adult man to whom marriage is forbidden due to kinship, suckling, or affinity. It is not a condition that the mahram be the husband; a woman can choose another mahram. If the husband insists on being the sole mahram for his wife, it is advisable to accept that to avoid negative consequences.