No.
Speaking more generally. That warrior's life, from birth to death, becomes part of that bloodheritage's history. How they live their final days or die absolutely matters, their entire story (their codex) does at that point.
The ways in which someone can lose a bloodname (willingly or unwillingly) are extremely limited, basically being Abjured by Clan society as an individual, or your Clan being Abjured. The Wolves-in-Exile and Nova Cats defied the Abjurement and continued to use their bloodnames anyway, and rebuilt eugenics programs based on the bloodnames they took with them into exile (for the Exiled Wolves) or the bloodnames they hold exclusively (for the Nova Cats). But in the eyes of the other Clans for official purposes, they were treated like they didn't have bloodnames anymore. For example Clan Wolf filled all of the Exiled Wolves' bloodnames with new bloodnamed warriors just as if all of the Exiled Wolves had died the day they were abjured.
So to do what you are describing, the failed warrior in question would have to go truly rogue, bandit. I'm talking run away from the Clan and Clan society entirely. At that point they are a bandit and are fair game closer to the manner in which you've described (to be hunted down, if the opportunity presents itself). But also probably a target of an individual Abjurement and lose their bloodname anyway. Also at that point, all the accomplishments of their life are overshadowed by the fact that they ended their life a dishonorable bandit.
Finally, I'd just say, except for the Diamond Sharks/Sea Foxes. failed/test-down warriors aren't demoted to other castes. They are just moved to different positions within the warrior caste. They become sibko instructors, or test pilots, or consultants working with lower castes on military projects, or Watch operatives. Or they get transferred to something like jumpship/dropship officer assignments. Or some other warrior caste assignment that's necessary but more of a support role. If they are/were high ranking, they may be made a military governor over a territory the Clan controls. We know of one example, Cyrilla Ward, who was effectively retired from active duty but still served her Clan as a much-respected Bloodname House Leader until she committed suicide so that she could nominate Phelan for the Trial of Bloodright for her bloodname.
Finally and most obviously, even for warriors who take on some of the above jobs first, they end their lives in solahma units made up of conventional infantry or old vehicles, which aren't even assigned medics. It's the last stop, the last assignment, a last chance to die in combat before their bodies completely break down. The Clans, at least in the Homeworlds, didn't tolerate the idea of retirement homes. Among the lower castes in the Homeworlds if you didn't work at all, you starved. The Clans didn't waste resources on unproductive people.
So that's a fate worst than dying in combat, to be avoided. Even by a test-down warrior who has sought to extend their lifespan as long as possible all these years.
I'd say for a young no-bloodname warrior, to go after a bloodnamed warrior to try and end their life for no other obvious reason than to make that bloodname available, would be seen as dishonorable. Other Bloodnamed Warriors would find that kind of behavior cringeworthy (aka the Khans, Loremaster, every member of the Clan Council, your own Bloodname House leader) would try to discourage this. For example, they might band together and decide the killer won't get a nomination for that Trial of Bloodright. They might also declare that given that warrior's motives, their actions were dishonorable, and in turn arrange for someone to challenge that warrior to a Trial of Grievance over the issue. See where this is going?
