When Sauron corrupted the Númenoreans during Ar-Pharazôn's reign by making them worship Melkor, were the King and his followers aware that Melkor, the Lord of Darkness, was in fact the evil Morgoth who afflicted their ancestors in the First Age?

If so, then how would Sauron have reinterpreted the early history of Arda to sway these people?

Most likely, but it wouldn't have mattered to him

Ar-Pharazôn would have known that:

1. Morgoth was the enemy of the Valar during the First Age.

2. Sauron was a lieutenant of Morgoth during that time.

3. Sauron was urging Ar-Pharazôn to worship a being named Melkor and referred to him as his lord.

4. Sauron described Melkor as being an enemy of the Valar.

Given that, I believe it's very likely that Ar-Pharazôn made the connection between Melkor and Morgoth.

Sauron didn't have to reinterpret history, but only to feed into the Númenóreans' distrust and resentment of the Valar that had been building up for generations.

Here are some quotes from AKALLABÊTH that demonstrate the Númenóreans' attitudes towards the Valar (these are in chronological order so you can see the development):

Now this yearning grew ever greater with the years; and the Númenóreans began to hunger for the undying city that they saw from afar, and the desire of everlasting life, to escape from death and the ending of delight, grew strong upon them; and ever as their power and glory grew greater their unquiet increased.

...

Thus it was that a shadow fell upon them: in which maybe the will of Morgoth was at work that still moved in the world. And the Númenóreans began to murmur, at first in their hearts, and then in open words, against the doom of Men, and most of all against the Ban which forbade them to sail into the West.

...

But Atanamir [king of Númenór at the time] was ill pleased with the counsel of the Messengers [of the Valar] and gave little heed to it, and the greater part of his people followed him; for they wished still to escape death in their own day, not waiting upon hope.

...

In those days the Shadow grew deeper upon Númenor; and the lives of the Kings of the House of Elros waned because of their rebellion, but they hardened their hearts the more against the Valar.

...

Great was the anger of Ar-Pharazôn at these tidings, and as he pondered long in secret, his heart was filled with the desire of power unbounded and the sole dominion of his will. And he determined without counsel of the Valar, or the aid of any wisdom but his own, that the title of King of Men he would himself claim...

In summary, the Númenóreans -- particularly their kings and the King's Men party -- resented their mortality and their prohibition from from settling Valinor. When the Valar told them it was for their own good, they weren't satisfied and continued to grow in their distrust of them.

So Ar-Pharazôn wouldn’t have cared that Melkor / Morgoth fought against the Valar in the First Age. Even if he considered that Morgoth fought against his ancestors, he would have likely thought that his ancestors were deceived by the Valar and had been fighting on the wrong side.