Why incorporate a business in Italy instead of Germany?
It is clear and confirmed you will remain a tax resident in Germany in nay case. Hence you just keep declaring and paying your taxes as before, in Germany. If you want to incorporate a business do it in Germany, keep your funds on your German bank accounts and pay from those accounts/cc your expenses in Italy.
In Italy I would stay under the radar, don't move too much money and don't declare anything nor incorporate any business, even you stay there over 6 months. If this situation is temporary and for no more than 2︀ years, chances that Italian tax authorities get aware of anything are negligible.
If you plan︁ to continue doing this for a longer period it's another story and rather complex one.︂
As some have mentioned, even you believe you declare your income correctly in both countries︃ and use DTT as guidance, tax authorities might still go after you and demand you︄ to pay the tax they believe you own them + interest + fines.
From then︅ on it's legal battle against the tax authorities that have unlimited funding and time. Don't︆ believe tax authorities play the game correct or fair, hence a good tax lawyer (that︇ will cost you 400-1000 eur per hour) will be required.
So keep quiet in Italy︈ and after two years make sure you don't spend too much time there anymore (or︉ move your family to there) in the following two years you stayed there. Keep your︊ residence anywhere (banks, id, ..) registered in Germany.