Zoumu (same as Zouyume but keepin' it onyomi :P [JPN for elephant and dream]) Chanozou (JPN for tea elephant) Kenzou (wise elephant)
...keeps smooshing JPN words together...
Nominating "Xiangu"
Xiang = Chinese for elephant, also means fragrant Xiang gu = Chinese for shiitake mushroom He Xiangu = Taoist deity who carries a lotus flower

Aura Calmer Tusker Lorath Chill Quietude
Yalakora
Maybe Gajhant? It's a mix of "Gaja", a word with Hindi/Buddhist origins that describes several positive attributes like abundance, strength and wisdom and "phant" from elephant, because I'm that original.
(It felt cool because Buddhist tradition says Buddha came to be in his mother's womb in the form of an elephant with six tusks and I totally just took this info off Wikipedia.)

The pokemon in all of us is coming thru xD
Chashin - tea god Ikimono - creature
Manfred ( cuz ice age) Mannoth (also kind of movie reference) Furrus
I feel accomplished thank u I think I'm done submitting LMAO I spammed too much
Elephantea because I am lame. Otherwise there are lots of great suggestions already. Give me a small woodland critter to name and I can do it...Elephant...not so sure. haha
Pyrrthea
Ganali (gahn-ahlee) Lungaii (loon-guy) Eraku (era-koo or eh-rah-koo) Eyaru (e-yahroo) Raizou (rye-zow) Shiang (she-ahng) Yala (Yah-lah also, an M.I.A song. Play on YOLO, because reincarnation, you always live always.)
Yusdremra
I fell like it is just a calming name, really flows off the tongue.
. . .
Nazozo
nazo - mystery z? - Elephant japanese
My personal favorite is Elololololol. This calls for an elephant bag like your bunny bag, but with the description of Elololololol.
Name: Oneiropaki
Reason: Oneiro is the greek root for dream, and paki is similar to the pachy in pachyderm. Also this gives the Ontra some company in the O part of the pet-phabet.
Name: Kangiden-cha
Reason: a pretty transparent ripoff of a Japanese version of Ganesha, there is wisdom which is typical Ganesha aspect, but also duality of love and dream of love in the two-headed version. Unlike many versions of Ganesha which are fairly discretely elephant-ish, this one is often painted,drawn or depicted in a misty flowing style which will go well with the flowing mane as described. Plus cha for Tea
Name: Phantisattva
Reason: phant from elephant natch, and sattva is often a suffix for those arhats whose buddha nature is so strong but has led them to come back again and help people achieve Nirvana instead of reaching the Nirvana themselves, which they could do if they wanted. This name version is more sansrkit-seeming so it also goes well with the already existing ancient-India theme of already released CSC goodies and the old costume trunk.
Zephyraphant
Zephyr
RIP Mom, I love & miss you more than you know. Tell Dede hi.
Ocean Conservation Namibia on YouTube... they do good work! https://www.ocnamibia.org