Knowing what I know now, I would probably build something like this a different way, and I don’t know if you can make this change or not, but I thought I would suggest it. Dealing with using data from different sites in a multisite setup is always difficult.
Anyway, this is what I would do.
1) get rid of the taxonomy field
2) Use a basic select field (or checkbox, radio field)
3) Populate the choices of this field dynamically (term->term_id => term->term_name) https://www.advancedcustomfields.com/resources/dynamically-populate-a-select-fields-choices/
This solution lets you avoid the problems you’re facing with the taxonomy field.
