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My sense is that there are at least three issues here, all important but needing to be disentangled and addressed individually.

"a more clear cut objective approach to categorization"

See The "objectivity"/"subjectivity" spectrum: having one's cake and eating it too. "Objective" reflects the (appropriate) aspiration to have things about which there is widespread "subjective" agreement. It is not either a value in its own right nor something fully achievable. We may find consensual understandings of rocks, and of measles that prove useful. Ought we to expect to find them about the states of mind of individual humans? Or is that perhaps something where, to one degree or another, we need to acknowledge that consensual understandings are necessarily limited in their usefulness?

"the major problem in my mind with trying to classify mental illness in terms of behavior and emotion is how subjective things are"

There is a long history in psychology of both starting with the "subjective" (gestalt psychology, phenomenology) and of trying to do away with it entirely (behaviorism). And an interesting recent history in cogntive psychology and neuroscience of begining to accept the necessity to admit as legitimate observations "subjective" reports (in, for example, studies of the neural underpinnings of pain, of emotions, and of consciousness). Maybe its time to generally recognize/accept that "subjective" things are relevant to "objective" inquiries? Maybe we can have our cake and eat it too?

"What provides us with a more useful categorical model? While each perspective may offer different understandings, which should be top priority?"

Why choose between the two? Or give one "top prority"? How about using both "structures, organization, chemical structures, etc" AND subjective reports, and letting usefulness in particular contexts determine the balance?