I have found finn's lute long before I ever got the quest.

I had to lug it around for ages, before I came to Solitude. Here, I talked to the right person, got my reward, but the lute's not removed. It's still a quest item and cannot be dropped.

I found a console option to remove the lute from my inventory, but even then, the carry weight is not corrected.
Is there a further console code to correct my carry weight for the (now missing) lute?

It's a quest item, so its displayed weight is immaterial.

Ah, thanks. Didn't know that.