Interesting topic, thank you for tagging me, Simi. I'm no mytho or Mahabharata expert but with my limited knowledge, I'll put down my PoV.
I think to answer this question, 'deprivation' needs to be defined first & also compared between the two of them, because we can only say 'more deprived' or 'less deprived' in relative terms.
Is it just the 'throne' we're talking about?
Then Bheeshma's sacrifice was greater because he could have had it & let it go, but Karna was never eligible for it like others have already said.
Is it having a family?
Then again Bheeshma is the more deprived one because he was robbed of the love of a wife & children because of his oath.
Is it identity? Status? Acceptance?
Then obviously Karna is the more deprived one.
Bheeshma's oath was a choice. It was a sacrifice, yes, that he made for the love of his father, but he made it. No one forced him to. So if he's deprived, then he's self-deprived, which seems like the bigger deprivation. But depriving himself of the throne did not lose him any other privileges. He was still revered and remained the King's chief Adviser. And the most significant, to my mind, is the fact that he never regretted his choice, nor was he ever bitter about giving up the throne and choosing celibacy. He also never confessed to missing having his own family because his half-brothers were his own as were their children and their children. He was 'Pitamaha' to everyone. His social standing wasn’t affected by his oath-induced deprivation, if anything he got an even more exalted status because of it.
Karna, on the other hand, had deprivation thrust upon him by circumstances. He may not have been eligible for the throne, true, but even so, his entire identity was erased due to Kunti's impulsiveness. He lost his social standing, and all his life faced humiliation & taunts of being low born that hurt him. He was deeply embittered by it and therefore latched on to the one person, Duryodhana, who embraced him despite his low-born status and not just that, uplifted him from that status. He bought Karna's life-long loyalty with that gesture. So that status mattered to Karna a great deal. He may have had riches & the joys of a family but his true identity/ status was denied to him all his life.
In this instance, it appears that Karna was more 'deprived'.
They were both Kshatriyas and to identify as one & take pride in those duties mattered more than anything else, so I think going with this aspect, Karna was the more deprived one.
Edited by LizzieBennet - 3 months ago
I have not gone much details but as far as i know bheeshma was responsible for Amba's miseries. Whatever he faced it was because he also did wrong with some one else and we can consider it as his karma bite him back. Pandava's respectd bhishma even though he was not a king and the decision was taken by bheeshma himself and he was adult.so he did not faced humiliation as much as karna has faced. 
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