Can I change the input of my reductionduring the proof?
No, you may not modify $I_1$. By the time you get to Step 3, everything is determined and you might not change your instances. Say $f$ is the polytime transformation from Step 2. Then you need to show

$I_1$ is a yes-instance of $\Pi_1$ if and only if $f(I_1) = I_2$ is a yes-instance of $\Pi_2$.

If after arriving at Step 3 you notice that your $I_2$ instance does not allow you to prove the statement, then you might want to change $f$ in your Step 2, and redo Step 3.