In Tibetan Buddhism, there is no single fixed form of The Four Immeasurables Aspirational Prayer. There are countless variations that interpret and expand the basic list of four items. They are not even always listed in the same order.
In your passage above the order is: Loving-Kindness, Compassion, Joy, Equanimity. So the phrase you're asking about corresponds to Joy.
Other variations give different interpretations of Joy. For example this one:
How wonderful it would be if all sentient beings were never parted from upper rebirth and liberation’s excellent bliss. May they never be parted. I shall cause them never to be parted. Guru Chenrezig, please inspire me to be able to do so.
As you can see here instead of "the supreme happiness which is without suffering" they speak about "upper rebirth and liberation’s excellent bliss". In both cases the emphasis is on "never be parted" or "never be dissociated".
I've seen other variations that speak along the same lines. The basic idea is clear: the Immeasurable Joy is what we cultivate towards those who already have some measure of happiness, either the worldly happiness, or the spiritual happiness of encountering the true Dharma, or the pure happiness of Supreme Enlightenment.
The emphasis is definitely on whatever happiness they have, regardless of how much or how "refined" it is. These beings may be somewhat healthy, live in somewhat comfortable conditions, somewhat not poor, somewhat without a conflict, somewhat educated, somewhat familiar with Dharma etc. - whatever happiness they may have we are happy on their behalf and wish them to never lose that happiness and ideally even increase it, up until "liberation’s excellent bliss".