What grammatical classification would one place a word created by ending a famous person's name with -ing?

In this context, Juno is being used as a verb. Since the inherit meaning of the "to Juno" is "to do something Juno is known for", then Junoing is the gerund form of the verb "to Juno".

Consider the word "Google" which inherently is a noun or adjective (The name of a major tech company (noun), a number (noun and adj) (specifically 10 to the 100th power or 1 followed by 100 Zeroes). However, when one searches for something on the internet, they are "googling" something, taken from taken from the tech company's name, as it is there most famous product line's service.

While "Juno" is ordinary a proper noun, you are using it in context as a verb with a unique meaning.

In the example sentence, "Anna was Junoing..." Junoing is not a gerund. It's the (invented) verb Juno used in the past progressive tense.

Compare:

Anna Junos as if gasps are just as essential as the air she breathed => present
Anna is Junoing as if gasps are just as essential as the air she breathed => present progressive
Anna has Junoed as if gasps were just as essential as the air she breathed=> present perfect
Anna Junoed as if gasps were just as essential as the air she breathed => simple past
Anna was Junoing as if gasps were just as essential as the air she breathed => past progressive

To be a gerund, a word that's typically a verb must be used in a way typically reserved for nouns.

Books (noun) are my favorite things
Reading (gerund) is my favorite thing
Junoing is my favorite thing

Anna was Junoing as if gasps were just as essential as the air she breathed.

"Junoing" in this sentence is a verb used in the progressive/continuous structure "Be [verb]-ing".

A verb form ending in -ing is traditionally classified as a present participle when it occurs in this structure.*

The -ing form of verbs in English developed from several different historical forms.* This conflation means that it is not entirely straightforward to categorize -ing words as participles vs. gerunds. But in one tradition of English grammatical terminology, the term "gerund" is reserved for -ing forms that are used in positions where noun (phrase)s would be used: for example, as the subject of a clause, or as the object of a preposition. "Junoing" is not used in such a position in this sentence, so according to that tradition, it would not be called a "gerund".

A more recent term belonging to a newer analysis of English grammar is "gerund-participle". This term is used by the Cambridge Grammar of the English Language, by Huddleston and Pullum, published in 2002, a source that does not recognize a categorial distinction between present participles and gerunds.

*The historical development of the progressive/continuous construction seems to be unclear: the -ing form could have been influenced from both an older construction involving a present participle and a construction involving a gerund. See this question and its answers for more details on this point: progressive forms: participle or gerund?