Is it possible that it's simply a communal process that was performed by a kohein but really wasn't part of kodshim?
That wouldn't be dependent on whether there is meila or not. Kodshim Kalim are not subject to meilah until their blood is sprinkled on the mizbeach even though they are definitely part of kodshim.
The Rambam writes Me'ila 2:5
פָּרָה אֲדֻמָּה מוֹעֲלִין בָּהּ מִשֶּׁהֻקְדְּשָׁה עַד שֶׁתֵּעָשֶׂה אֵפֶר. אַף עַל פִּי שֶׁהִיא כְּקָדְשֵׁי בֶּדֶק הַבַּיִת הֲרֵי נֶאֱמַר בָּהּ (במדבר יט ט) "חַטָּאת הִיא". וּתְנַאי בֵּית דִּין הוּא שֶׁלֹּא יִמְעל אָדָם בְּאֵפֶר הַפָּרָה:
The prohibition against me'ilah applies to the Para Aduma from the time it was consecrated until it was reduced to ash. Even though its status is that of an entity consecrated for the improvement of the Temple, concerning it, Numbers 19:9 states: "It is a sin-offering." Beis Din made a condition that a person does not violate the prohibition against me'ilah with the ashes of the red heifer.
A proof that it is NOT a part of Kodshim but rather a communal process to obtain purity is that the Rambam places Hilchos Para Aduma in the halachas of Tahara and not in the laws concerning Avoda in the Beis Hamikdush.