The indefatigable marlene has opend the discussion on another German poet who is of great interest to me, too. I will chime in after trying to get hold of the essay Die Wunde Heine by T. W. Adorno, who should have much to say about this. (Not that I take Adorno as final authority on many subjects, but I find him really illuminating when it comes to literary criticism because of his penchant for debunking established wisdom)
BTW Die Wunde Heine is hard to translate. It means literally "The Wound Heine", which isn't really English. One has a choice between "Heine as [Open] Wound", "Heine's Wound", "Heine, the [Open] Wound" and perhaps other possibilities. I probably like the first one best.
Addendum: After starting this thread, I have translated two Heine poems I like. They may offer a good way to get into this topic.
"Bare-handed speech synthesis"
10 hours ago