Let's say more years go by and they do decide, "we need a generational ship".
Rather than the Captain ordering them to commence breeding I think she would do the math and decide they needed X amount of children. She would then implore/woo all females of child bearing age (which in Trek times is apparently puberty to 55+) to consider volunteering. They would have the option of breeding the traditional way or via insemination, this is assuming they hadn't figured out parthenogenesis yet though that might not be optimal in this situation. Some might be willing to have quite a few ala Phlox's wife and they would be on much lighter duties, in fact child raising, teaching and ultimately training in ship running would become important positions.
The Captain of course would not be bearing any young.