Net value of saving a child's life from a negative utilitarian perspective?

Answering this or similar questions will be challenging for any worldview that takes into account second-order and long-run consequences of actions, not just negative utilitarianism.

Saving a child has many such effects that will be very difficult to account for: not just effects on loved ones but also effects on the ecosystem, climate change, demand for meat, the economy more generally, etc. So assessing the grief experienced by loved ones is probably only a small piece of the answer to your overall question. At the same time, it might be particularly salient or important because the bond is personal and irreplaceable. If this life is not saved, we can do little to offset that harm.

For what it’s worth, a negative utilitarian theory might also include the frustration of preferences in the evaluation of an action. To the extent that the child wants to continue living, this would provide reasons to save them, even by negative utilitarian lights. Whether this is a decisive reason is another matter of course. 

If you do find negative utilitarianism or other suffering-focused views compelling, I think it makes more sense to ask the question: according to this view, what could be the very best thing I could be doing with my time and money? Most people who have asked this question have come up with interventions that seem much more impactful than saving lives directly -- regardless of whether the latter would overall be a good thing. Here is one person's attempt to answer this very difficult question: https://reducing-suffering.org/

How I think of the impact of saving a life (by donating to the likes of AMF):

• a life is saved, and the grief caused by that death is averted
• the person whose life is saved lives the rest of their life
• Total fertility rates reduce because of lower child mortality
• In terms of total number of lives lived, the saving-lives effect and the reducing-fertility rates effect probably roughly cancel each other out in places were the current fertility is high (source: David Roodman on GiveWell blog)

So saving the life helps us, one life at a time, to transition to a world where people have fewer children and are able to invest more in each of them (and averts plenty of bereavement grief along the way)

I am glad you are seriously considering the implications of your philosophical beliefs -- this is laudable. I very much hope you don't conclude it's bad to save children's lives.