And can't the same sensation be either 'happy' or 'unhappy' depending on the circumstances? A person with persistent sexual arousal syndrome isn't made 'happy' by the orgasms they can't help but 'endure'.

Yes, this is true. We will need to assume that the button can analyze the context to determine how to provide happiness for the particular brain it's attached to.

My point is that happiness is not necessarily associated with accomplishment or objective improvement in oneself (though it can be). In such a situation, some people might not value this kind of detached happiness, but that doesn't mean it's not happiness.