Vrish,
Whatever he says would be deemed a lie. May be he could have said "I do NOT know my caste" - isn't that a lie too ! The problem his he doesn't know his birth for a fact.
There is never half truth - no fuzzy logic :(
It is quite unfortunate he was put in that situation none other than fate (in other words Kunti/Suryadev). Role of mother/father is clearly delineated in smrithi.
Anyways , back to Parashurama - if you consider him as GOD, He would have known his identity NOT through a bee piercing Karn, but by divya dhristi. It is divine destiny to have him cursed. So there needs to be some reason for curse/his death, that's pretty much it.
The greatness of Karn is he not only stood against all odds, but the charity he has done without even caring for his life is something far far superior. All he wanted was his talent to be recognized. As I said a tiger can be a tiger, you cannot tell him to be a cat
He never got his birth right the way he ought to have got, but he got more than all that after death - everlasting glory.