But ultimately, the only way I find to cope is not caring.

It's important to distinguish between emotions and decision theory. You can (try to) be perfectly altruistic in calculated decisions, while not caring on an emotional level. Better, you can care in more positive ways: feel good when you help, but don't feel guilty for not helping, or feel painfully strong empathy for the suffering, except to the extent that doing so actually motivates you sustainably. You aren't obligated to feel any emotion that doesn't win.