The beauty of this type of project is the fact different writers can have different interpretation. This character is a tool. A doll to play with, and different people can play in different ways. For example, in the Doctors of the Church hub, Bright IS just an amulet without a host, being worshipped by cultist, and gets annoyed when revived, which is an interesting way of fitting him into this canon.
I understand that this "he's an anomaly, but also works for the Foundation" business is a slippery slope. Technically, yes, it's not impossible that the organization would keep him employed after a work-related accident, and I tend to give the older personnel more leeway in this regard, but this should be used cautiously. If used at all.
And well, you can write Bright like you've suggested. Even if at some point he was allowed to retain his position, something could happen that would take this opportunity away. What if everyone got sick of his shit and decided he's too big of a liability? Or what if O5-6 decided it would be for the best, showing his "brotherly love" in this perverted kind of way?
Or what if the opposite happened: Bright does not want to keep getting revived, but he's being made to, because he's too much of an asset. Almost like Clef who's being prevented from retiring by blackmail, 166 and her safety being used as leverage. Wouldn't be unrealistic, either.