According to Macmillan Dictionary, definition #1:
used for talking about what was going to happen in the past
used for showing what someone expected, intended, promised etc when they were thinking or talking about the future:
James said he would never forgive her.
Most analysts expected that there would be a change in policy.
Consequently, if the context has the reference point in the past, as in your example, we would use would when we were thinking or talking about the future and the sentence reads as:
but other ladies carried such things, so she also would have to carry one.
On the other hand, if we use had to have one as you suggested, the sentence reads:
but other ladies carried such things, so she also had to have one.
To me, both versions imply that she's obliged, bound or under the necessity of carrying one, that is, two different ways of saying the same thing.