Two questions regarding SN 46:54

First, what is “the deliverance of the beautiful?” I haven’t encountered the phrase anywhere else in the suttas and can’t recall any positive discussion of beauty as something to be cultivated.

And how, bhikkhus, is the liberation of the mind by lovingkindness developed? What does it have as its destination, its culmination, its fruit, its final goal? Here, bhikkhus, a bhikkhu develops the enlightenment factor of mindfulness accompanied by lovingkindness … the enlightenment factor of equanimity accompanied by lovingkindness, based upon seclusion, dispassion, and cessation, maturing in release. If he wishes: ‘May I dwell perceiving the repulsive in the unrepulsive,’ he dwells perceiving the repulsive therein. If he wishes: ‘May I dwell perceiving the unrepulsive in the repulsive,’ he dwells perceiving the unrepulsive therein. If he wishes: ‘May I dwell perceiving the repulsive in the unrepulsive and in the repulsive,’ he dwells perceiving the repulsive therein. If he wishes: ‘May I dwell perceiving the unrepulsive in the repulsive and in the unrepulsive,’ he dwells perceiving the unrepulsive therein. If he wishes: ‘Avoiding both the unrepulsive and the repulsive, may I dwell equanimously, mindful and clearly comprehending,’ then he dwells therein equanimously, mindful and clearly comprehending. Or else he enters and dwells in the deliverance of the beautiful. Bhikkhus, the liberation of mind by lovingkindness has the beautiful as its culmination, I say, for a wise bhikkhu here who has not penetrated to a superior liberation.

Second, the sutta makes associations between the bramhaviharas and the arupa jhanas. How should I understand these associations? As one example:

“And how, bhikkhus, is the liberation of the mind by compassion developed? What does it have as its destination, its culmination, its fruit, its final goal? Here, bhikkhus, a bhikkhu develops the enlightenment factor of mindfulness accompanied by compassion … the enlightenment factor of equanimity accompanied by compassion, based upon seclusion, dispassion, and cessation, maturing in release. If he wishes: ‘May I dwell perceiving the repulsive in the unrepulsive,’ he dwells perceiving the repulsive therein…. If he wishes: ‘Avoiding both the unrepulsive and the repulsive, may I dwell equanimously, mindful and clearly comprehending,’ then he dwells therein equanimously, mindful and clearly comprehending. Or else, with the complete transcendence of perceptions of forms, with the passing away of perceptions of sensory impingement, with nonattention to perceptions of diversity, aware that ‘space is infinite,’ he enters and dwells in the base of the infinity of space. Bhikkhus, the liberation of mind by compassion has the base of the infinity of space as its culmination, I say, for a wise bhikkhu here who has not penetrated to a superior liberation.

  1. The beautiful literally means beauty, and as pointed out in SN 46.54 this is preceded by and on condition of the ability to control sensuality. This is connected with the roads to power. Perception control is discussed in “The Roots of Good and Evil,” Nyanaponika:

"These fives modes thus constitute a subtle “magic of
transformation” by which pleasant and unpleasant feelings,
as they habitually arise, can be changed at will or replaced
by equanimity. A mind that has gone through this training
has passed the most severe test, indeed. Through that
training, it obtains an increasing control over emotive
reactions, and internal independence from the influence of
habits and passions. It is said in the Satipaṭṭhāna Sutta, “He
dwells independent and clings to nothing.”

This begins with developing the strategies of Majhima Nikaya 20.

  1. The spatial quality of the brahma viharas is connected with the non-material states, and Analayo has made a beginning on this in “Compassion and Emptiness.”

Knowledge of form precedes and is a condition for the formless:

"DN 15: “Possessed of form, one sees forms. This is the first emancipation. Not percipient of form internally, one sees forms externally. This is the second emancipation. One is intent only on the beautiful. This is the third emancipation.”

Here’s Bhante Sujato’s translation of SN46.54:

The apex of the heart’s release by love is the beautiful, I say, for a mendicant who has not penetrated to a higher freedom.
Subhaparamāhaṁ, bhikkhave, mettācetovimuttiṁ vadāmi, idhapaññassa bhikkhuno uttarivimuttiṁ appaṭivijjhato.
SuttaCentral

Subha , the word used in the sutta for “beauty” also means auspicious and welfare.
So the lines could refer to these aspects as well – the auspiciousness of a mind liberated by compassion; the welfare of a mind liberated…

Can you clarify about the “spatial quality of the brahma viharas?” Is this referring to the common refrain about pervading the different directions with eg mudita?

Yes, ‘boundless’ refers to space, more broadly the whole path can be known in terms of development of perception of the element space:

"What if I — overpowering the world [of the five senses] and having determined my mind — were to dwell with an awareness that was abundant & enlarged? Having done so, these evil, unskillful mental states — greed, ill will, & contentiousness — would not come into being. With their abandoning, my mind would become unlimited, immeasurable, & well developed.'—Majhima Nikaya 106

You might find this topic useful: The apex of the heart’s release by love is the beautiful - Discussion - Discuss & Discover (suttacentral.net)