If not, what is preventing "happiness" from being quantified, in principle?
Quantifying whether someone is happy is not the same as being able to stimulate the happy feelings. The experience machine is critical to your question:
Nozick asks us to imagine a machine that could give us whatever desirable or pleasurable experiences we could want. Psychologists have figured out a way to stimulate a person's brain to induce pleasurable experiences that the subject could not distinguish from those he would have apart from the machine. He then asks, if given the choice, would we prefer the machine to real life?
Nozick also believes that if pleasure were the only intrinsic value, people would have an overriding reason to be hooked up to an "experience machine," which would produce favorable sensations.
It turns out that when Nozick asked people if they'd like being hooked up to an 'experience machine', most people said no. I found a random poll which has 38% of people wanting to plug in and 62% wanting real life. Note that this is just a forums poll, so take it with a grain of salt.
Assuming 'no' to the experience machine: We are not guaranteed that human thriving has finite 'definition'. Consider an analogous question: will science ever 'finish'? Will we ever discover every physical law, with the only possible research left to be discovering the initial conditions of the universe to finer and finer precision? It's not clear. Using induction, we could say that we ought to expect science to continue finding deeper and deeper laws of the universe. Perhaps human thriving is the same: so far we seem to be finding more and more complex forms of human thriving; who says it will ever cease? Indeed, we know from addiction studies that just about any routine stimulation of the brain loses its effectiveness in creating fulfillment, over time. Our brains seem to get bored of things and want new experiences.
Given this, I would say that no, we will be very bad at predicting future happiness, just like we're bad at predicting future science, unless you're talking only one or two years into the future. We can make very broad predictions—like more greenery makes for more happiness—but there is no evidence that we'll be able to be precise in the way that Sam Harris seems to require.