Native Spanish speaker here (ES-MX, specifically, if it matters). I think this is one of the cases where a solid general rule breaks down in the specifics.
You are correct about the difference between "ser" (to be, permanently/over an indeterminate time) and "estar" (to be in a particular state right now). But "No soy feliz" sounds perfectly idiomatic to me, even for a relatively transient state of sadness. ("No estoy feliz" doesn't sound wrong to me either, but feels just slightly less natural than "No soy feliz" even in a context like "No soy feliz ahorita", with an explicit "right now").
As a note: "No estoy contento" (Also "I am not happy", or maybe "I am not in a good mood") is definitely "estoy", rather than "soy". No clue why "No soy feliz" does feel idiomatic.
You are correct about the difference between "ser" (to be, permanently/over an indeterminate time) and "estar" (to be in a particular state right now). But "No soy feliz" sounds perfectly idiomatic to me, even for a relatively transient state of sadness. ("No estoy feliz" doesn't sound wrong to me either, but feels just slightly less natural than "No soy feliz" even in a context like "No soy feliz ahorita", with an explicit "right now").
As a note: "No estoy contento" (Also "I am not happy", or maybe "I am not in a good mood") is definitely "estoy", rather than "soy". No clue why "No soy feliz" does feel idiomatic.