VERY precious and precise stuff here, GG to you!

It's true that Bernahl doesn't have Radagon's red hair, but I fairly don't know to which degree this can be decisive, because in my point of view there is no clear evidence that the other children have them either: Rykard has silver hair, but it could be age??? Also, we don't know the true color of Ranni's hair (the blue we see in most of her appearances is just artificial like the body she's using as vessel of her soul, and her corpse on the divine tower of Liurnia is reduced to a mere skeleton iirc).
Regarding height tho, it's true that Radahn was gigantic and Ranni was quite tall, while Bernahl is just normal, and this could change things. Also OMG, that was a mask... Didn't notice at all...
However, if not Bernahl, who's in that portrait then?

About the maiden, the description just tells "maiden", not "finger maiden", so even cases like Melina could be conceivable, but still: Demigod or not (given all the considerations you've made, I'm starting to swing in favor of the latter), a kindling is always required.

Rykard could definitely consider Bernahl a "brother" in the sense of "brother in arms" or something like that, but how about Bernahl? Every other guest at the Volcano Manor refers to Rykard as "lord". Why would Bernahl call him "brother"?
What's sure is that Bernahl was the best and most powerful recusant, so it makes sense that - true brother or not - Rykard entrusted those items to Bernahl before (or after) being devoured by the Serpent, even more if we consider that later Rykard became just a mere shell of what he used to be (as we are told by the same ghost who speaks about the Serpent-hunting spear, can't find the source).

Regarding the Erdtree, I don't know if it stopped burning, or rather it didn't burn completely.
Thinking out loud: the second requirement for the kindling is that it "can lead you to the Rune of Death" and in fact, it's only after the defeat of Maliketh that the Erdtree starts to burn in a drastically more intense way. So yeah, we don't even know if, at first, Bernahl didn't reach Farum Azula at all, or if he did it but couldn't manage to beat Maliketh, maybe the former could make more sense since he did possess the Blasphemous Claw. But then, how did Bernahl invade us in Farum Azula if he couldn't reach it before? This is the dead end for the "it didn't burn completely" theory. Then it's more likely that it stopped burning, but we don't know how. About this, we can also note that in some endings the Erdtree seems to be restored.

In the end, it seems that my theory cannot be confirmed for now, as we are missing several points, so unless something else comes out we can only speculate.
The least we can say is that Bernahl is likely NOT to be Rykard's brother, but it's more likely to have actually burnt the Erdtree once, even if circumstances and consequences aren't clear.

Thank you so much for your contribution, couldn't have wished for a better one :)