There's (short of FAI) always another child to save.

This is literally untrue. According to Wolfram Alpha, there are 1.855 billion children on earth (2009 estimate). Of course, not all of them need saving. Giving What We Can cites the official estimates from the United Nations Development Program for the [edited for clarity: annual] cost of basic interventions:

Basic education for all: $6 billion
Water and sanitation for all: $9 billion
Basic health and nutrition: $13 billion

These are finite numbers for which humanity doesn't need FAI. Of course, they are too huge for any one individual. So normal-earning individual donors cannot, in fact, be consistent about donating only a part of their disposable income and spend another part on luxury assuming they want to be perfect utilitarians. But it makes this condition false:

But if all states of the world - whether they be donating 0%, 10%, or 20% of income - result in 0% total goal fulfillment

Numerically, feeding all human children on earth - and providing family planning to all potential parents - is clearly feasible without FAI.

In order to do the most good for our fellow humans

For what it's worth, some of us consider non-human sentients as relevant too.