Well by the time Constantine established Constantinople his capital, Rome had already ceased to be the capital some time earlier.

The functional capital was wherever the Emperor happened to be living at the time. And by the crisis of the third century these functional capitals more often than not were on or near the border, so the Emperor could respond to barbarian incursions. In 293 Maximian divided the empire into quarters. The Gaulic capital was Trier. The Italian capital was Mediolanium (Milan). The Illyrian capital was Sirmium. The Baltic/Asian capital was Nicomedia. You may note that all of these were positioned close to the Rhine and Danube frontiers. You may also note that Rome wasn't on that list. Certainly Rome was still the biggest city. And it had a lot of powerful and influential men. But what it didn't contain was an emperor or the imperial bureaucracy.

When Constantine established Constantinople, what he was really doing was moving the Eastern capital from Nicomedia, which was just across the Bosporus. Constantinople was a stronger and better defensive position. It also had a better harbor. For similar reasons the Western capital was moved from Milan to Ravenna in 402. So no one really had a problem finding Constantinople. It was visible from the old Eastern capital.

And Imperial infrastructure wasn't all that at the time anyway. It is only later that what was becoming the Byzantine empire developed the intensely robust bureaucracy. The earlier Empire tended to be more decentralized with a lot of authority devolved to the local authorities. As power centralized in Constantinople, the palace complex grew to accommodate it.

How was the move communicated to the many other kingdoms with which Rome was acquainted?

I think you might be presupposing that capital cities work like they do today where formal embassies are all established and the government stays in one spot. Back then, envoys would have to deal with the emperor for serious diplomacy and he could be anywhere, but he would be in a known location (his palace, on a hunt, on campaign against some tribe or another).

To what degree did Byzantium possess the infrastructure to accommodate the world's largest imperial power?

It didn't, and that wasn't what Constantine was after. Although Constantine ended up as sole ruler, he began as a tetrarch and ended by dividing the empire up amongst his sons because it was far too large for one ruler alone to manage. Constantinople as an administrative hub developed more under the Byzantine period in a much reduced Empire.

A very good book on this is "Three Christian Capitals: Topography and Politics" By Richard Krautheimer. Full text of it can be found here: https://archive.org/details/bub_gb_-Rk0fpRIKLAC

In 324, Constantinople already was not laid out like any of the dozens of other cities founded by previous or later emperors or by Constantine himself. Nor was it an imperial residence set up within an older town and provided with the imperial paraphernalia — palace, circus, huge thermae — such as the tetrarchic capitals Trier, Milan, Thessalonike, and Nicomcdia. .. it was an alternative capital, a second Rome, equal but not superior to the old Rome, to quote Constantine's dedication decree.

The book starts with a section on Rome, and then moves on to the founding of Constantinople. There it deals with both the building projects themselves, and Constantine's motivations in building them.