@TylerLawson, I spent a long time trying to make sense of your comment https://chat.stackexchange.com/transcript/message/56727087#56727087.
I couldn't quite prove the first fiber sequence you wrote - but I'm realizing maybe I'm misinterpreting what you meant by pointed/unpointed stabilization - so wanted to write it a bit more carefully:
To simplify the setup, I'm just gonna consider the identity functor, and hence it's sufficient to consider stabilization functors (and not the derivatives).