Kurn wasn't a "normal" Klingon. I doubt "normal" Klingons would consider him a Klingon at all thanks to the repercussions of Worf's actions and Gowron publicly stripping the Mogh of House of everything. It's reasonable to ask whether in Kurn's case if his fate is actually worse than suicide?
I meant normal klingon relative to worf. As in he wasn't raised by humans, he didn't grow up in a human world with human views and beliefs influencing his development.
He grew up a klingon, in a klingon world, with normal klingon beliefs... And I think it's fairly easy to assume that a klingon with that background and belief system would absolutely hate the idea of a memory wipe/ false identity. They'd probably consider it a dishonorable lie as well as the cowardly way out.