I want to say something like, ‘Would many women, if asked, say they feel unsatisfied with the way they are treated? They would say that they /do/.’
A friend advised me to change ‘do’ to ‘did’. But I’m not so sure…
did is the past tense of do.
If you say “They would say that they do” then this is the present tense which matches “feel” in the question and implies that the dissatisfaction is ongoing.
If you say “They would say that they did” then this is the past tense which does not match and it implies that they were dissatisfied in the past, which is in conflict with “… are treated”. They cannot be dissatisfied in the past with something that is happening now.
So I suggest that you are right and your friend is wrong.
However it is very common to hear did in spoken English even though it it not strictly grammatically correct.