After taking my little girl to some photo sessions in studios when she was 1mo, 2mo, 3mo, 6mo, I have some opinion about it (from the consumer side, just to give a perspective from who is paying for it):
1 - My little girl wasn't crying before the flash and began to cry after you shoot it: it's your fault. Doesn't matter if it will harm her little eyes or not. She didn't like all that light on her face, neither did I.
2 - You have all those lenses with all those speeds and apertures and everything else: choose the ones that will require less light. You can do it.
3 - The room doesn't need to be dark, with light flashing just at the photos. Put some soft light on it
4 - Sometimes you don't need to put that much light, we want to see her beautiful smile, not screen her for skin defects and so on.
5 - Side, back, top lights can create some funny shadows and won't bother her that much.
Of course, bounce it, use soft-boxes, put it at the minimum essential. Be creative, play with the child/baby, make her find it funny when you make some sound, some funny face, and that light appears. Give time so that the little one is confident that that thing (=light, sound) won't do any harm.
Ask parents to help you with that. For example, my little one always look at me or at my wife when something unexpected happens. If we just smile, or play with her, the 2nd or 3rd she won't even bother seeking for a signal of "should I cry or not?".
As you can see, the "will flash hurt his/her eyes" is just a small part of everything. A piece of some other answer gives a good point: not only make good pictures, make it a good experience, a professional time, and give some care to the parents, too. You'll find that in the end (=in the second half of the studio time) you'll get wonderful pictures, if you spent the first half gaining confidence.