Can't say I fully understand the Brunhilde - Loki interaction here. Why did Brun ask Loki specifically? And why did Loki accept?
I can only guess about it myself, but as I understand it, it was something like this:
Given how hard the situation currently is, the chances of Humanity winning are quite small for a very simple reason:
They CAN'T beat Odin.
So they have to win the other 2 fights, and for that, they need to be ABSOLUTELY SURE of who'll fight on each one.
Loki knew that, so when Brunhilde called for him, he thought that she may have given up, and was going to give up officially and beg Loki to at least spare her remaining sisters and herself.
Especially, since apparently they have history together, even if Brunhilde doesn't return his feelings, winning now would mean killing him, something that he didn't think Brunhildr would actually be willing to do.
When Brunhildr begs him to be the next combatant, that's completely thrown out the window.
Brunhildr is determined to win, and since they can't beat Odin, that means killing him, sending against him the only fighter that has an actual chance of doing it.
This moment probably hits Loki like a truck, since while he's used and maybe accepts that his beloved Brunhildr won't return his feelings, he was not expecting that she would actually want to kill him.
So, he hides his pain behind laughs and lies, while fulfilling the petition of the woman he loves and changing the roster to become the next fighter, and then going to his room to cry in solitude, trying to drown it in laughter before coming to the resolution that now, it's given Brunhildr's choice, one of them will have to die.
And he won't let himself be the one to die.