While all the stated options are technically implementable, these are severely limited by the piece graphics available to me from Eric's 1-kanji set and from Ludii's Extended Shogi set. Many of the pieces have the same names but different moves when compared to their Shogi counterparts. However, I will summarize my thoughts on your picks below.
Flying Ox: While their are other options for this piece, none really go well with the theme of the games, which is a drops-playable version of their Shogi counterpart. Then there's also the problem of the Free Boar and what to change that to (this piece has fewer options and the same problem).
Golden Bird and Heavenly Horse: The former's full Kanji name translates to "Oriental greenfinch," and the first Kanji in the latter's full Kanji name also translates to "winged." Under this logic, you could call them Golden Finch and Winged Horse, respectively, and it would only barely be cheating.
Kirin, Phoenix, Lion, and Treacherous Fox: Unfortunately, these don't really have options for alternate names within the limits that I set for myself. For the latter, there is another fox piece that I could use (Dai Dai Shogi's Enchanted Fox), but it is displayed incorrectly in Ludii, thus making it unviable.
The Greenwade icons for the Strong Bear and Running Leopard may need to be adjusted depending on your preferences.
While all the stated options are technically implementable, these are severely limited by the piece graphics available to me from Eric's 1-kanji set and from Ludii's Extended Shogi set. Many of the pieces have the same names but different moves when compared to their Shogi counterparts. However, I will summarize my thoughts on your picks below.
The Greenwade icons for the Strong Bear and Running Leopard may need to be adjusted depending on your preferences.