So: does the intonation of the harmonic actually change when I release it,
Because the pressure of the bow on the string changes the tension (and, microscopically, the length) of the string. It's not dissimilar to a guitarist who bends pitch by sliding the string laterally on the fretboard. I suppose that the action of the bow driving the string's vibration has an additional impact on the pitch, but I'm less sure of that.
When I was in grad school, one of the professors, a lutenist, was very vocal about string players who play lightly when they're tuning their instruments. He said that they should play at full volume when they tune because otherwise they would not be tuning the actual pitches they would be playing in the performance.
If they are different, should I try to tune the harmonics based on the bowed note? Or should I try to tune to the released note?
If you're playing mostly with the bow, yes, tune the pitch you hear when you're bowing. If you're playing mostly pizzicato, it probably makes more sense to tune the pitch you hear when you let the string vibrate freely.