Angels, Demons, and the souls of mortals are immortal, and cannot truly be destroyed (except by The One or by all 21 gods acting in concert; only Mulcarn has ever really been destroyed). They can only be banished to the vault of the god whom they served to rest and recover, or to Arawn's underworld if no one claims them. Demons tend to have short term goals so when they kill angels the angels are just kept out of the picture for a few centuries, but when Basium kills demons he imprisons their souls to until judgment day when The One would (in Basium's estimation) damn them for all eternity.
Odio was not killed. He was imprisoned by Kilmorph after she defended Aeron remove his ara of protection from him. This was after the gods and their archangels (excluding Odio, who refused to stop his rampage through Kilmorph's lands to attend the meeting) met at The Seven Pines, but before they had reached any agreement as to what The Compact should say.
Basium showed up for the meeting, where he probably argued that the Godswar should end with evil's unconditional surrender. He stormed out of the meeting and left Arawn's service after most of The Compact had been written, but before it was signed. I'm pretty sure Arawn did withdraw his Ara from Basium, but that Basium has not (at least yet) done anything so severe as let a god act out against him like Kilmorph did to Odio without violating the Compact in the process. (Basium and the Mercurians have spent most of their time fighting demons on their own turf, in Hell, where The Compact doesn't apply.) It is possible that Kilmorph's action against Odio wouldn't even be permitted, but it happened before The Compact was signed so there was nothing stopping her then.
As we know from Sphener's pedia entry, the Compact clearly allows angels to fall, leaving their god's service and protection and becoming essentially mortals (although probably still not aging or being vulnerable to diseases and the like). From the Compact's point of view, Basium, Cassiel, and Sphener are free agents, whose actions cannot qualify as breaches of the Compact but who don't get any special protections for it either.
As they are essentially considered to be mortals now, their souls would go to Arawn unless someone claims them. Basium's essence is still closely tied to the precept of Life, which could be called the sphere of Persistence. Sucellus does seem more eager to collect souls than the other good gods do (probably because he has the smallest army) and he would have a valid claim of Basium, but Basium hates Sucellus intensely and would never agree to serve him if he did find himself in his vault (assuming Nemed made a vault that Sucellus now inhabits and that Sucellus doesn't just hover in spirit form accross Erebus) he would probably do all he could to try to usurp his precept. That could be a path for him to turn evil, but I don't think Sucellus would give him that opportunity. It is likely that Basium would return to the Netherworld, not because he was made by Arawn but because he does not have a loyalty to another god. However, since Basium once guarded the passages from the world of the dead back to the way of the living, he could easily navigate his way back out to Erebus, even if it means that he must be born in a weaker human form since he was not willing to wait for his divine body to regenerate. The evil gods try to claim every soul they possibly can, even if their claims are quite weak, as they believe their great machine can turn the most good souls evil given enough time. Camulos would probably use Basium's violence and rage against demons to try to claim his soul, but whether this would succeed is unknown. Basium would surely continue fighting demons there, but that is fine with Camulos as everyone in his hell spends all his time fighting everyone else anyway. This is a hell of pure violence so constant and extreme that it in time makes its inhabitants forget why they fight. It is quite possibly that Basium would seek to usurp Camulos's precept since he has found Sucellus' out if his grasp. He would only want this power to fight demons, but he may forget that in time. Everyone in Camulos' sphere, especially his archangel and strongest true demons, is trying to usurp the precept, so Baisum wouldn't be all that special. It is also possible that the evil gods would claim that Basium's desire to return to fighting like in the godswar constitutes a connection to the Ice sphere, and thus mean he should be destined to that hell mostly for those without his determination. Considering that this sphere is without a god and that the last claimant to that position is now being held in the Netherworld of the Netherblade and will likely soon be a slave of Larth, Basium may see this as an easy way get the power he needs to fight demons. I could see him becoming the God of Ice, and promptly breaking the Compact just as Mulcarn did. I can't really see Basium ever progressing to Aeron's or Agares vaults, but there is a slim chance he could become the God of Chaos or the God of Ice, in which case he would surely be corrupted and become quite evil. His godhood would not be good for humanity, but he would probably still fight the other evil gods and would break the great machine of hell.