The sentence is correct in its original form. Your second suggestion gives a wrong parsing. Your first suggestion is at least not wrong, probably also fully correct but nobody would actually say it like that. I’m going to offer a fourth version modified from your first which works, too. But one thing after another.
Stripping away everything that is not required to form a complete sentence, the shortest form is:
Die Tage haben Spuren hinterlassen.
Spuren being the accusative object to the verb hinterlassen which is in perfect tense and die Tage being the sentences subject. So far, so clear. And there should be no question about in Bourges — an adverbial of location — or wenigen — an adjective to the subject noun Tage — either. What remains is how to parse Sterne-Ruhm. It belongs to the days, it clarifies which days we are talking about. If you wanted, you could consider it an attribute to the days in which case it should techinically be in genitive case — hence why your first suggestion seems correct to me (without me being able to make a clear call). However, it would be more common to use such a genitive case with an article, hence my fourth variant:
In Bourges haben die wenigen Tage des Sterne-Ruhms Spuren hinterlassen.
I suspect that something happened here that is also known from the genitive case that some prepositions would generally require: namely that the ‘complicated’ genitive has been replaced by a nominative construction. Compare:
Wegen Schneefall fällt heute der Unterricht aus.
Here, too, genitive would be formally correct but it is not as common in my experience; instead a replacement case is used which happens to be dative here, as can be seen using a plural form which labels dative:
Wegen Unwettern fällt der Unterricht aus
This also allows us to determine that Sterne-Ruhm is not in dative case by choosing a different noun and applying plural:
Die wenigen Tage Unwetter haben Spuren hinterlassen.
And finally, why is your second suggestion wrong? Well, Spuren is part of the object while Sterne-Ruhm belongs to the subject. They are not part of the same sentence fragment and thus cannot be a single word. If needed, we can reorder the sentence to show this:
Spuren haben die wenigen Tage Sterne-Ruhm in Bourges hinterlassen.