Fair warning: In the areas where the rules aren't 100% comprehensive such as this, I try to adjust my interpretations to maximize the aspects of the game that my groups prefer. That is, we try to make it as logical and as simple as possible, but in a pinch we'll sacrifice logic for simplicity and ease/speed of game play.
For LST's, we clearly don't want them to be especially easier or harder to take down simply because of how they're affiliated. It should impact their effectiveness, (i.e. their dice pools) but it shouldn't be shifting their stress tracks around. So your options B and C are right out. We don't want silly cases where Thanos buddies up with a Skreet and that causes him to faint dead away, or he gets punched out by Howard the Duck because he was in Buddy affiliation and a d8 stressed him out. For this reason, I always use the largest Affiliation dice pool as the "stress" track. For most LSTs, this is the Solo Affiliation.
When an LST is creating dice pools, I use whichever Affiliation the LST is in. So if Thanos is in Buddy affiliation, his Affiliation die is a d6, but he's still taking stress against the 3d10 (from his Solo Affiliation). This can present situations where the LST is down to one die of stress, but because they're in a Team affiliation they're getting multiple (typically smaller) dice. Unlike your D case, we don't degrade other affiliations that may have more dice than the LST has left in their Solo affiliation. That's a little counterintuitive, since we're used to the LST getting weaker as we hammer on them, and in this case we're not reducing the Affiliation die in use, so we don't get that advantage. However, in most cases this makes some sense narratively, because the Team or Buddy dice represent the LST using other characters to manage the situation, so the effect of their own degraded status is minimized. Plus it's a little piece of paperwork that we don't have to track that doesn't usually have a major impact anyway.
Lastly, in a sense we treat the multiple Affiliation dice as individual stress tracks. That is, I can physically stress out one of his dice, emotionally stress out a second, complicate out a third, and so on until I take him down. Different types of stress or complications that don't exceed the die size can stick just like they would on a normal character. So Thanos can have a d8 of physical stress, a d6 of mental stress, a d8 of emotional stress, and a d10 Enraged complication all sitting on him for attackers to pick and choose from when opposing him. But if that complication gets pushed to a d12 or higher, it takes out one of his 3d10 Solo affiliation dice dropping him to 2d10, and the Complication goes with it.
We use A, but we count all damage against the highest affiliation, regardless of which affiliation they are in and using for their rolls. Each die can be stressed or complicated out by any type of damage that exceeds it. The LST doesn't go down until each of those dice are stressed or complicated out. And we don't discount the other Affiliations. Even when he's down to 1d10 in Solo, in Team he still gets to use 2d8.