This is indeed a tricky challenge, and you've articulated it well. However, without knowing the missing element ("brand name" / "mother name"), it's hard for me to give you specific guidance.
Does the brand name itself suggest health and wellness (as, for example, "Hygieia Wellness" might), or is it arbitrary (as with Oscar Health, the insurance company) or even fanciful/coined (e.g., AbbVie)? Each of these directions might dictate a different naming architecture.
I'd also want to know how the sub-brands will be identified and promoted. How much interplay will there be among the five verticals?
Finally, how were the five verticals identified under the old company name? Is there an advantage to maintaining the prior sub-brand naming system?
Without that knowledge, I can only comment that "Wellness" has general, umbrella-brand connotations and does not sound "too much like just another one of the five verticals," which are quite narrowly defined.
One last thing: I would caution you and your client about #5 ("health"), which does have a semantic overlap with "wellness." Perhaps this vertical deserves a more clearly commerce-related name -- XYZ Wellness Store, for example.
Happy to chat on a call if you'd like to go into this further!