What is the aerial equivalent of bathing when one enjoys the breeze?

Bathing would ordinarily be a good translation, and poetic because of alliteration with balmy breeze, but you probably want a different verb since you’re using bathe to contrast with the first part of the translation.

Basking is another good word for bathing in the air, and it’s also alliterative, but it has connotations of warmth that you may not want in this specific context.

Cooling directly expresses the sensation of relaxing in a breeze. Chilling is also possible, but literally it would imply some discomfort; there’s a slang meaning of “relaxing,” but it’s probably the wrong register for poetry.

To bathe (or soak) in the Yi river, to enjoy the caress of breezes at the Wuyu altar, to return home reciting poetry.

Perhaps you should consider a term you already used - wash

to pour, sweep, or flow in a stream or current: waves of pioneers washing westward

The term is often used to indicate a flow of almost anything. This ngram shows usages for breezed washed and air washed.

Perhaps

To bathe in the Yi river, to wash in the breezes by the Wuyu altar, to sing back home.