@KFrank
Thank you for your feedback!  That was my initial reaction as well, but between posting this and reading your response, I went ahead and tried training with a None label added.  The new model seemed to do quite well.  The accuracy of my other labels seemed improved if anything (at least as far as I’m able to interpret accuracy at this point).  Maybe I’m just thinking about it wrong since, technically, that last label is “Recovered at Home” (which I DO actually want to be able to predict) rather than “None”, but that still begs the question about how label interdependencies might influence model results.  (NOTE: I realized after posting that there’s another obvious inter-label relation present in my use case, namely you won’t be intubated if you’re not hospitalized.)
My use case is also, to put it mildly, atypical.  In the end, the likelihoods themselves are the critical output, not a hard label prediction.  I still plan to do some threshold analysis and possibly renormalize my  likelihoods, but the ultimate goal is to have the ability to predict “risk of an outcome” rather than the outcome itself.  Does any of that change anything?  Or would you still recommend removing the final label and trying to figure out a method to estimate the “Recovered at Home” aka “None” outcome after the fact?
Thanks again for taking the time!
–SEH