He is maximizing his utility part-function: people should not be harmed. If I know of a situation such that: Someone will be harmed if I do nothing; They will be harmed equally if I tell the authorities; If I intercede with my allies, they will be harmed less.
The first two actions are equal. The third one is better. The ability to determine the actual consequences of future actions is magical. Fortunately for Harry, he has access to magic. I expect Harry to develop a procedure where he makes several plans, checks for a note indicating which plan he should use and any changes he should make, executes the plan, and then time-turns a note back to himself indicating which plan worked and any retroactive changes to it, then assists himself if needed.
Something tells me that the note would be more likely to say something like "DO NOT MESS WITH TIME".
r/HPMOR readers on heroic responsibility - not the OP, the comments. Holy snorkels this is good.