What would that have accomplished?
The eastern frontier of the Holy Roman Empire pretty much remained stagnant after the Kingdom of Bohemia lost Upper Silesia in 1469.
Were the Hungarian and Czech nobility willing to join the Holy Roman Empire? The decision isn't solely that of the monarch - maybe in theory, but not in practice.
Something that's never being clear to me is why Hungary, Croatia and other Hapsburg domains in the Balkans and that, were never made part of the HRE before it's collapse.
Was their any particular reason or did they just never get around to doing it at all?
The Czech lands (Bohemia-Moravia) were already in the HRE, so the Czech nobility didn't have to join, they were already there.
Just about Charlemagne. The Franks were a Germanic nation, but he cannot be truly be called a Frenchman or a German. And he was born in West Francia according to probabilities (Francia's capital was Paris).The HRE was always seen predominately as a "German" empire despite the Roman name. Even when northern Italy was a part of the Empire it was partly based on historical events of German tribal groups that had set up kingdoms and then Charlemagne (who was a German, not a Frenchman!). Even the Czechs did not take up much of Bohemia/Moravia back then, Germans were dominant in a lot of what is today the Czech Republic; the Czechs back then were heavily Germanized in culture as well. The Habsburgs never tried to reform the HRE by adding in Hungary (which you'd have to in order to add Croatia, Croatia was subservient to the Hungarian Crown, not the Austrian) because Germans would never have allowed it.
Now, a good AH story could be made about Habsburgs some how getting the British after the Napoleonic Wars to allow them to throw off Hungary as independent in return for Bavaria, Wurtemburg, Baden and Saxony and therefore be stronger against Prussia. Would that allow the Habsburgs to unite Germany, if even using alien bats to force the British into this POD.