I think it very clear that the both of you are approaching this question from very different angles.
Odola, you are wanting to emphasize and justify Inglor being "of the House of Finrod", and the connection to Gildor - thus the suggestions to show a young Gildor at the leavetaking, and to have Finrod declare, officially, that his companions in death are his kin.
Nick, you are wanting to work on story structure, where elements are included as they make sense for those characters and those circumstances. You don't want there to be 'distractions' arising during the moments leading up to Finrod's sacrifice and death. Time out for a technicality while people are being eaten by wolves doesn't seem to fit.
I think one possible solution would be to deal with this matter much earlier. In the very beginning of Episode 6, when it's just Finrod, Beren, and the 10 companions on their way...it would be appropriate for Finrod to say a few words thanking them for their loyalty in accompanying him, as well as warning them that this quest is likely to end in death for all of them (or perhaps more euphemistically about their chances of return from Angband). If he does this before they even encounter the orcs, it won't distract from what is to come, and it serves more as a last-chance-to-back-out than an attempt to memorialize people who are actively dying.
Would this help?
This would imho leave out Finrod's dismay at people of Nargothrond and his kin in particular doing nothing to get them out. The C-bros were all for retaking Tol Sirion but now that Finrod is held captive there they hold Orodreth back. This does amount to a betrayal. Luthien is on her way to get Beren out even before he is captured and nothing holds her back. (I know Finrod does not know this - but the audience does and that is enough for the story). Orodreth is held back by the C-bros' few words. Finrod has all the right in the world to feel bitter.
I also think to put Finrod's declaration at the beginning of their journey would be compromising Finrod's call to all of his people demanding their loyalty in the abdication scene. Reminding his companions shortly afterwards that they could die - after most of Nargothrond sides with the C-bros for exactly the same reason - seems unnecessary. One always "can die" on a mission to Angband - even as an elf. But there is a difference between the mere possibility of dying and actualy being eaten alive by werewolves while those in duty to rescue you do not move a finger.
So for me there is a natural theme of loyalty in Finrod's story. First his kingdom betrays him - then his family. As such his reaction in that very moment makes sense.
I do not really see what a generic vague reminder about the possibilty of dying before leave-taking would bring to the story and how it would free Finrod to be unconcerned enough to just discuss philosophy with Beren while his companions are being killed off one by one for their loyalty to their leader?
What other strong and story-economic reaction can you give Finrod in the dungeons instead?
The companions' death is on Finrod. He decides to bring them along, to go via Tol Sirion, he has lost the song battle.
He could have chosen to go with Beren alone or to take another route. It was not their oath to Barahir after all, just his. As such he does owe them and has to yet pay them back while he is alive.
Finrod's fate is to die fulfilling his oath, but not necessary to treat his own companions as disposable extras?
The more underlying question is - do we tell an athropo-centric story where Beren - just by being the human hero - is the centre of the story's universe and all others - Luthien included, are just props, objects? He is the protagonist - the other's just mere tools and catalists for his action and can be ignored once their task is done?
Or are we telling a story where elves are also real characters and e.g. their deaths do matter?
I was under the impression that it is the latter.
As such the dungeon is imho hardly a place to make the story solely Beren-centric again - he is not the one to die there.
Imho it does not really lessen Beren as a character if we and the audience take a moment in the dungeon to honour the fallen...
It really is not always just all about him and his love interest.
But I see also how the impulse to make the story Beren-centric at this very moment might overrule all that. The story as told by Tolkien is Beren-centric, Finrod's companions and Finrod's death itself serve in the texts only the purpose to elevate Beren's status in the story - as that of the fated hero.
I just do not think this attitude serves the perspective from which our story is told, where "elves are real people, too". And it lessens Finrod as a character in the stories that came before, as a person and as a leader imho.
I have stated my case. Do with it as you want.