What about the possibility of the HRE becoming an HREIN instead of an HREGN. That is, that the HRE sheds German territory and power, focussing on Italy, instead of the other way around.
The only reasonable way would be a the formation of a unitary kingdom of Italy before Otto traipses down from the Alps to claim the crown of Rex Romanorum and Emperor: a successful Berengar (or in alternative one of the Guidonids of Spoleto). Otto stays in Germany, with the royal crown, but does not recognise the imperial crown claimed by Berengar. Over time, the imperial crown survives but just as a title of the King of Italy (or possibly Italy and Arles) and Germany goes on as a kingdom.What about the possibility of the HRE becoming an HREIN instead of an HREGN. That is, that the HRE sheds German territory and power, focussing on Italy, instead of the other way around.
Berengar is not a well known historical figure, also because not a lot of primary sources survived. Calling him the "cockroach King" however may be right on the money. He had to face daunting odds (the Magyars, the Saracens, foreign and domestic claimants to his crown, unruly and treacherous feudatories) and was defeated more than once. And every time he survived, and came back stronger.
Brenta
I have trouble with the notion that Brenta was winnable for Berengar. In the first place, this is Berengar we're talking about - if he ought to be famous for anything, it's for remaining in power for decades despite losing several critical battles. His record is not one of military accomplishment.
More importantly, however, Lechfeld was the culmination of decades of learning, preparation, and construction on the part of the German kings and Bavarian dukes. Over the course of the first half of the 10th century, the Germans built up a formidable system of defense-in-depth which allowed them to harry the Magyars as they returned from plundering, and trained a cavalry force with the proper tactics to defeat the Magyars. These were hard-won lessons, learned in spilled blood and lost treasure. The true accomplishment of Lechfeld was not the actual battle, which was somewhat inconclusive (the Magyars were forced to withdraw, but the Germans suffered heavy losses), but the several days after the battle in which Bavarian militia/local forces destroyed the Magyars in detail at river crossings and fords across the duchy.
I think it was quite possible for the Italians to beat the Magyars, and maybe even for Berengar to manage it - but not the first time they met. I don't think anyone beat the Magyars on their first encounter. Berengar might have been a lucky man, but to miraculously defeat an enemy he was totally unfamiliar with and whom the Germans took half a century to adapt to is too lucky even for him. His best bet was probably to not fight Brenta at all, but then again, letting the Magyars run unchecked through Lombardy was not going to help his support and legitimacy. To be honest I'm not really sure what Berengar could have done to salvage that situation. (And it's worth noting, by the way, that even Brenta didn't cripple him - he still ruled Italy for another 25 years afterwards.)
Berengar's lack of a son is indeed a big problem. I agree that the Anscarids are probably the best option; their large territory in the north was always a thorn in the side of Berengar and later Hugh, and making the Margrave of Ivrea also King of Italy is probably the best way to make Lombardy politically stable. If Berengar manages to hang on and stay alive for another 5 years or so, he probably outlives his son-in-law, and by 930 he can pass his kingdom directly to his ~30 year old grandson Berengar II.
I also agree re: expansion - the (post-Lombard, pre-Ottonian) Italian kings were much more interested in Burgundy than southern Italy. Firstly, many of them were themselves Burgundinians; but Burgundy was also part of "Middle Francia" which, under Lothair, had been attached to the Italian and imperial crowns. You don't really need a direct marital connection to justify such a conquest (though it always helps). I think Arles/Provence/Burgundy is much more likely an acquisition (or attempted acquisition) than southern Italy at that time, though if the Kingdom of (nothern) Italy thrives, I'm sure they would eventually get involved in the south.
I have to disagree a bit on the importance of Lechfeld: by the mid X century the Magyar invasions were already decreasing, because the German lords had learnt better tactics to cope with them certainly but also because more and more Magyars had abandoned a nomad lifestyle. For political reasons the victory was touted around as the end of the Magyar danger, same as the victory of Charles Martel at Poitiers had been depicted as the "saving of Christian Europe". There were good spin doctors in the Middle Age too, or maybe I'm too much of a cynical soul.
There are obvious ties between Lombardy and Burgundy, and not just dinasty care ones. In an age when the seas were made dangerous by the Vikings in the north and by the Saracens in the Mediterranean a lot of commerce was transiting from Flanders to Lombardy and vice versa through the Gotthard pass. However I'm not necessarily proposing a dinasty union between Arles/Burgundy and Italy. I would be perfectly satisfied by a recognized commonality of interests, chief among them the need to keep West Frankia from becoming too powerful and East Frankia from intervening either in Burgundy or Italy. In my opinion just keeping southern France independent from the Royal power in Paris and friendly to both Burgundy and Italy would be a very successful outcome.