So your hypothesis is that Harry will win by doing the thing he already tried and failed to do, and got a stern warning from the Universe for trying. The one thing that he can't do, that's the thing he'll do.

Okay, except you've not done the work. If Eliezer puts a huge road block in front of an obvious solution to the story — "NOPE, can't do that, sorry" — and your hypothesis is "No he'll do it anyway", then the actual work is not just saying he'll do it (since the story explicitly states the insane power Harry would have if it did work; it's not like that's a big discovery itself) but rather saying how we get from our current state of impossibility to the state where Harry pulls it off.

"If Harry can figure out how to bypass the limitations of the time turner" — that dismissive 'If' is the entire problem you should be trying to solve.