I know a Jewish guy in the States whose named Michael but is called Micah - I think it might be a family nickname or his Jewish name, not sure which really.
And an Israeli co-worker of my Dh changed her name last year from Aziza to Mika (she says MEE-ka)...no idea why but Mica/Mika seems to be relatively popular there.
I think Micha and Michi are both cute - reminds me of a friend named Michon (no idea where her folks got that one)...perhaps Michel nn Micha/Michi? That's one of the the Czech spellings and I think it would get you a pronunciation a bit like Mitchell without the harshness which would lead nicely into Micha/Michi. Any Czech ancestry? LOL
I'm not a fan of Mike either but like Mick...all the Mick's I know have great senses of humour and the dirtiest laughs ;o) The problem with Micheal, even without the fadas, is that the aspirated C of Irish is something that you'll (and later he'll) always be explaining. It's just not done in English. You can go around fighting that ('Okay, try to say c the regular way -kayl, now say it without letting your tongue touch the roof of your mouth, -hyayl, Mee-hyayl' - would get tiring very shortly wouldn't it) or you can do what Irish guys living abroad the world over have found easier - try for the correct vowel sound while accepting the English C or go by a nn. In a more unusual name you have half a chance at imprinting a pronunciation but with a classic like Michael, it's just going to be an uphill battle all the way which is why I don't even go there.
We give all of ours ethnic family names...in the middle. It's too bloody irritating to go through where it's from, how it's said, how it's spelt, etc. all the time. There are 7 of them and 2 of us, time is at a premium, I have no patience, especially with nosey fools. So there's Sandor (SHAN-dor), Iosefo (yo-SAYF-o), Mariska (ma-RISH-ka), Elisapeta (eh-lees-a-PET-a), Zoltan, Aladar (say what you see with those) and Kai (KIGH like high) but you have to be in the inner sanctum to even know about them - Nic, George, Sam, Isaac, Liv, Jake and Phoebe are much easier to live with in our little Sussex world.
But Micha/Michi is sweet and straightforward really and I might consider honouring both grandfather's name and the German heritage in the English/German form of Michael with the German nn Micha/Michi.