Certainly, in early '54 being a Pilgrim was a big deal -- in the post war era Pilgrims were demonized "as treacherous, savage totalitarians", and they were blamed for many of Confed's failures during the Kilrathi war. The crew of the Tiger's Claw were quite normal to react with hatred to the fact that Blair flaunted his Pilgrim anscestry... but did they treat Paladin the same way? No -- because he didn't directly promote the fact that he was a Pilgrim... something that Blair presumably didn't do after the Tiger's Claw tour.
Of course, there are other reasons why it's no longer particulary important whether or not he's a Pilgrim... first, the acts of sabotage the Pilgrims were blamed for turned out to be the work of the Society of Mandarins, a far easier group to hate. Secondly, there *are* no more Pilgrims -- almost all of them leave the universe in Pilgrim Truth. Blair no longer represents a violent civil war, but is instead (if he even makes his anscestry known) an oddity... not to mention the fact that after the movie (and especially after WC1) he is regarded as a war hero or (after K'Tithrak Mang) a traitor. Even Gerald, with his understandable hatred of Pilgrims, reached an understanding with Blair for tihs reason.
(And if Blair didn't admit his heritage -- how would anyone know? He discovered what being a Pilgrim means, and he rejected that... it's doubtless that he has any interest in promoting his anscestry any more... and there are very few veterans of the Tiger's Claw to spread rumors).
The skipper missile is clearly a completely different weapon -- you've obviously listened to me talk about it before, but I can do so again if you'd like...