As I see it:
The thing is that there's nothing preventing the Staufen from addressing both Italian and German problems. Both are basically solvable with the right solutions.
The problem that is why I think this ultimately is overambitious is that the Staufen will have to periodically drop dealing with German problems to deal with Italian ones and vice-versa.
Which is why in my timeline Conrad IV chooses autonomous but tax-paying Italy to the campaigning necessary to make Imperial rule unquestioned, because said campaigning would get in the way of his administrative developments.
Its not that he can't launch those campaigns. Its not even that he can't win those campaigns. But in a decision between "setting up the foundations for Imperial bureaucracy", and "enforcing a greater level of control over the governance of the Italians", the former was chosen.
Any emperor in his shoes, or his father's, or his grandfather's, or his son's, will have to make that sort of choice - with consequences which range from relatively minor (Italy is still part of the Empire, it just has a greater level of self-rule with the Emperor intervening as overseer more than overlord) to things like how OTL Frederick II sacrificed the regalia for support now.
If this scenario considers an Italy that is part of the Empire but with considerable freedom as acceptable, it is doable. If it is necessary for Milan to be as independent of Imperial rule as Nuremburg, something will break. I don't know what something, and I don't know when. But that is too much.
The other side of things though is that there's no real reason the Italians will mind being part of the same state as the Germans or vice-versa. At least not in this period (and by the period it comes up, the empire will be well established).
What the Staufen have a problem with is (for instance) Milan wanting more autonomy to the point of being answerable to the Emperor in name only, if that, not the Kingdom of Italy being uniteable but refusing to merge with the Kingdom of Germany.
Hopefully this is as clear to the reader as it is to me. If anything looks weird, questionable, etc. please ask for clarification.