There are a few things worth considering here, so let's have a quick look at a couple of the big issues!

Edward III's relationship with the imperial throne began very soon after his birth in a widely circulated English prophecy, known as 'The Six Kings'. The prophecy foretold that Edward, as the mythical 'boar' with which King Arthur was associated, would wear 'three crowns'. It also predicted that he would eventually be laid to rest 'at Cologne amongst the tombs of the Three Kings'. As the triple crowns of the Holy Roman Empire are gold, silver and iron, Edward's elevation to the throne was expected by many through-out his life.

Edward's bond with the empire was perhaps closest in 1338, when the king met with emperor Ludwig in Koblenz on 5th September. They were joined by all of the imperial princes in procession to Koblenz cathedral, where Edward sat at the foot of the imperial throne. Ludvig then crowned Edward, in front of the assembled electors, as Imperial Vicar of the Holy Roman Empire. This conferred on the king of England the position of leader of the imperial allies in the Low Countries, as military campaigns were underway against the king of France. The German princes where therefore obliged to swear fealty to him.

Ludvig died on 11th October 1347 while hunting, which was a dangerous princely pursuit. When the electors met to choose his replacement, the name put forward was Edward's. Ludvig had previously stripped him of the Vicar's crown, but now Edward was offered the imperial throne (pending coronation by the pope). This was apparently the fulfillment of the prophecy so long associated with Edward. However, as you mentioned, he turned it down. Ian Mortimer, in his landmark work on Edward, gives us this insight:

'He turned the offer down. He no longer sought to add to his prestige through allusions to prophecies, or the acquisition of great titles. He no longer needed to associate himself with old kings and legends. His own reputation, won through his own efforts, and in new ways, was greatness enough in itself'.

Edward was a man deeply concerned with both his standing in a figurative sense and the exercise of his right in a physical sense. With his exploits in France he had elevated the kingdom of England to a position of diplomatic esteem unseen since the Angevin empire of his ancestors Henry II and Richard I. In this regard the crown of the Romans now mattered considerably less to him. Furthermore, as far as conquering France was concerned, he had already found the German princes and armies to be consistently improper and unreliable allies. Edward preferred his chances of securing the French throne with his own English troops, not those of the electors.

For more information on the issues of dynastic ties and diplomacy between England, the Empire and France see either:

Johnathon Sumption - The Hundred Years War (1999) Vol.1-3.

Anne Curry - Hundred Years War (2002)

For more information on Edward, the prophecy and his foreign policy, see:

Ian Mortimer - The Perfect King: The Life of Edward III, Father of the English Nation (2008)

W.M. Omerod - Edward III (2011)

(Edit: a bit of grammar)

1348 was also the year that the Black Plague arrived in England. In less than a year, over one third of the entire population of England was killed as a result of the plague. With the help of certain members of his administration, King Edward III did a pretty good job of keeping English society intact during the epidemic. I'm not sure why he would have turned down such an offer, but he would have had a lot on his hands during that time.

Also, some information about the Holy Roman Emperor during that time: The electors only elected the King of the Romans, until ~1519. The King of the Romans wasn't exactly the Holy Roman Emperor; the Holy Roman Emperor had to be declared and crowned by the Pope. While the King of the Romans would usually go on to become the Emperor, this wasn't always the case. During the time period you are asking about, Charles IV was elected to be the King of the Romans, not Edward III. Charles IV was actually elected to this position twice, in 1346 and 1349, but didn't become the Emperor until 1355. His predecessor, Emperor Louis IV, died in 1347, so it is possible that at some point in this time (1348), Edward III became involved, but I haven't seen anything regarding that.

edit:take a look at /u/eeeeeep's answer as well. Apparently the first part of my answer isn't wholly correct.

In addition to /u/oogachaka1 's reply, that particular period was a restless one in Germanic HRE lands. The Emperor Louis IV of Wittelsbach was busy consolidating his family's holdings, and in the process antagonized many german princes of the HRE. He gave Brandenburg to his eldest son when the prior ruling family died out. He gave Palatine to his newphews. Bohemia was taken as an imperial fief.

Lower and upper Bavaria was re-united into the Duchy of Bavaria under his own personal rule, in addition to Tyrol. Finally, he conferred several fiefdoms in the low countries to his wife Margaret of Holland. All this happened before 1348.

Even by 1346 the electors were already upset at Louis IV, and with the support of the Pope elected Charles IV of Luxembourg, a rival of Louis IV, to be King of the Romans.

Back to Edward III, he started out as an ally to Louis IV as they both were rivals to France. However, later on Louis IV broke the alliance and he himself tried to ally with France. In particular, it is important to note that Edward was married to Philippa of Hainault, so had claims that he later forwarded to the County of Hainaut. In the meantime, Louis IV had married as his second wife Margaret of Holland in 1324, giving him claims also to Hainaut and Holland.

In the end, Louis IV's side won the claims to Hainaut and Holland, after rivalry against Edward III of England.

So clearly in 1348 Louis IV and Edward III were in conflict, and Louis IV had agitated many of HRE's prince-electors. That said, I have never heard of Edward III being outright offered the crown as King of Romans. Can you share your source, please?

If he accepted it, does that mean England would have been part of the HRE?