'Phoenician', while morphologically the logical adjectival version of 'phoenix' (they are etymologically related), does not evoke the bird at all in modern usage.
'Phoenician' will most likely only evoke the ancient Mediterranean civilization, possibly their language,
or sometimes as a demonym for people from Phoenix, Arizona. Context would disambiguate these.
There are archaic/rare uses of 'phoenician' meaning bright red (like the bird) but the usual (still archaic/rare) term for that is
(both can be found in the OED).
Since you are creating labels for an invented thing, I'd recommend not using 'phoenician' because that is so strongly associated with the people. Using 'phoeniceous' will force people to think about what it means, and everyone will realize what you are trying to do (avoid the adjective for the people) and understand it as the adjective form for the bird.
One may think that, since many animal types get their adjectives (bovine for cows, feline for cats, etc), that bird types would get similar treatment. And they do. This veers off more into culture because all those '-ine' words were actual neologisms that happened to catch on. Similar terms for bird types were invented (anserine for geese, corvine for crow) but for the most part those haven't really caught on outside scientific terminology. Except for maybe:
which is metaphorical for the eagle's-beak shape of a nose.
Note that other birds get by with adding '-like': duck-like, goose-like, crow-like (if their scientific versions aren't available to you or your listeners). If you must follow this particular pattern, the word would be
but I personally find that infelicitous. 'Phoeniceous' feels better.