The big red button makes the fight roughly 10 seconds longer, mind you: the time it takes to run to the button and press it, as opposed to poking Mimiron. Any well geared 120 will one-shot each phase in hard-mode too.
I always activate it during my farm runs for an extra gold or two I get from vendoring the hard-mode chest.
But no, you should not need it for Mimiron’s Head.