Firstly you do know that also Philip the Good had similar negotiations with the emperor. Whereas both he and his son Charles the Bold were proud, Philip was a more realistic politician.
Yes, which he has to do given he wants a crown in imperial lands. But certainly not Frisia! He wants one of the Carolingian crowns; preferably Lotharingia. It's the most prestigious, the one that gives him the most de-jure claims, and the one that makes him most equal to France.
Again Friesland/Frisia was considered to be a historic kingdom and thus certainly an option. He, I assume you mean Charles the Bold, initially wanted to become king of the Romans, thus by Papal coronation become emperor and outrank the king of France. A territorial kingdom, considered in negotiations with both Philip the Good and Charles the Bold would be different. It would make grant them the same rank as the king of France, Denmark etc., but just like the kingdom of Bohemia, such a kingdom would be an imperial fief. Prestige comes from the higher rank, and as such resurrecting a 'historic' or historic kingdom would help.
Another point is granting most de jure claims, that exactly why the emperor is hesitant to do so, since the imperial princes, especially those involved won't be too pleased by that.
Which is part of why he has to ask; mind you, and yes, it would have been an important concession to make, I don't say otherwise. Which is why Burgundy wanted it. The Duke didn't want a king-title for the sake of having one, it had to have real meaning, real claims, and equivalence to France. He was strong enough to challenge the Emperor, his early demise notwithstanding, and had proven he believed in 'might makes right'. It's reasonable to believe the Emperor might have felt all of Frisia, Lotharingia and Burgundy could be lost to the Empire forever de-facto if he didn't compromise.
Regardless of the title, for any title inside the Empire, the emperor will be involved, often the Pope too. Recognized independent rulers outside the Empire could request a Crown from the Pope and/or Emperor. Except might didn't make everything right in this era, for instance they, well their lands, were wealthier than many kingdoms. In a sense diplomatic recognition makes right. I disagree on the being king bit, Philip in that regard was more realistic than Charles, who was susceptible for the symbolic value of being a king (in general). Charles by no means was being able to challenge the Emperor, maybe one on one, but that basically meant that Charles would have been surrounded by enemies, the Emperor and the king of France would under such a scenario join forces against him and he can't win from all his then enemies.
Still the Emperor wanted to bring the Low Countries (mostly, but not all, a part of the HRE) closer to the Empire again. One thing he did want to prevent at all cost, is France getting the entire Burgundian inheritance; though if they ally, the easiest deal would be to assign each their nominal fiefs.
At the time, he owns virtually all of it, and it seems like only a matter of time before the Netherlands are fully under his heel. But more importantly, Frisia is absolutely not as important a crown as Lotharingia or Burgundy. Those grant him legitimacy to rule up the Rhine, or as south as Arles, and give him solid claims on French-ruled land; they are inherently much better in addition to the fact that the ex-Carolingian crowns were still seen as a solid notch above the others, especially by the French.
That he owns a lot of it, is especially, why offering this historic Crown makes sense, since that is the deal most likely to be acceptable for the imperial princes and the Emperor, who needs to grant the crown. IIRC Philip the Good was granted the title imperial vicar west (left) of the Rhine. Lotharingia not only connects with Lorraine, but also with the legacy of Brabant & Limburg, the 'heirs' of Lower Lotharingia (just like Lorraine is the 'heir' of Upper Lotharingia). As I wrote earlier, the duke of Burgundy, as 'free' count palatine of Burgundy already held a fief of the kingdom of Burgundy (Arelat). Furthermore like the duchy of Saxony was reduced, when in was enfeoffed again to a new dynasty in 1180, so will probably also be a newly enfeoffed kingdom of Burgundy, certainly with respect to any claims on other imperial princes. In that regard Burgundy would work best with large nominal parts of that kingdom conquered by France. OTOH connecting the Lower (Low Countries) and Upper parts of the Burgundian Lands is more valuable, which would give more value to Lotharingia. However regardless which name, Lorraine could be made a vassal of that kingdom, though even that won't be a popular move and the emperor certainly won't transfer all vassal fiefs desired by Charles, but all fiefs already held by Philip or Charles certainly would have been a part of it.
Regarding Carolingian heritage, that too was important for Lotharingian 'heirs' like the duchies Brabant & Limburg (in personal union), but so was the Frisian kingdom in the counties of Holland & Zeeland (in personal union) and the duchy of Gelre (which had some Lotharingian claims too).
Anyway IMHO you're undervaluing the kingdom of Friesland/Frisia (as is my impression you probably think I overvalue it

).
I'm well aware! And I'm still convinced the Duke would have been offended if he had been offered Frisia. One of his strategic desires was to legitimize his expansionism in the Lorraine and Alsace, and as Duke of Burgundy, he did not see the Lowlands as the heart of his realm carved by the sword. He wanted to be on equal grounds with Paris. Frisia doesn't cut it. The reason why the Emperor was seriously entertaining the notion of granting him a crown in the first place was because of his de facto power; a form of appeasement, in large part. I don't think that if he had ultimately decided to go ahead with that, he would have risked going with Frisia. The Duke might have taken it if that's all that was on offer, but he would have been bitter about it; not very effective appeasement.
It all depends on which point during the negotiations, Charles started out unrealistically wanting to become king of the Romans (heir of the Empire). Philip for instance was less concerned about the title, but with any extra vassal fiefs attached to it. Charles seems to put it a bit anachronistically more 'Romantic', but if Friesland or Brabant would enable him to achieve most of his desires...
But the Low Countries with Flanders, Brabant and Holland, were the heart of his realm, since that's were most the money came from. I admit that it was a strategic desire of Burgundy to connect their Upper and Lower Lands. Ultimately deciding to carve things out by the sword is IMHO what was the final nail in his coffin, Charles overplayed his hand.
A crown of an imperial fief would have made him a king, so equal in rank, but not in prestige. Any historic crown Friesland, Burgundy or Lotharingia would have made it a bit less parvenu, which is something that would accompany a totally new kingdom. The latter is probably what the emperor initially had in mind.
I'd say approachment and not appeasement, the goal was to draw Burgundy closer to the empire not to appease the Great Duke of the West.
Of course, once the Emperor had decided NOT to do it, it made no sense to let records reflect that he even entertained the notion of parting with one of the Carolingian crowns. At that point, writing it was about Frisia makes entire sense for the Emperor, even if it's not true. And that explains why the sources differ.
We don't know, at various points and with different negotiations within two reigns various options were considered and we all know none came to fruiting. Not only sources differ, but in some cases one could sometimes wonder, if the cultural background of certain authors played a role too.
Another thing, which I noticed in our little discussion too, is point of view; what makes sense for the duke of Burgundy doesn't (always) apply to the Emperor and vice versa.