Verb tense agreement - 'Do' vs. 'did'

Solution 1:

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.

Solution 2:

I don't think "do" is the correct word to use.

Depending on context, either "did" or "are" is a better option.

If you're asking about a general feeling, use "are".

Would women say they feel unsatisfied? They would say that they are [unsatisfied].

If context means you're asking about how they felt about a specific interaction, then you would say...

Would many women say they felt unsatisfied with the customer service? They would say that they did [feel unsatisfied].

As pointed out below, it could also be:

Would many women say they feel unsatisfied with the customer service? They would say that they do [feel unsatisfied].

Frankly, I'd reorganize the sentence entirely to avoid the clunky phrasing by taking out "feel":

If asked, would many women say they are unsatisfied with the way they are treated?

Yes, they would.