Not really a suggestion but more of a request of how phoenix actually works and what items+potions have an effect and what doesn't work.
Is it correct that he doesn't attack, but just burns? And does attack speed and crit work on it?
That's right, phoenix does not attack so all on attack mods do not work on him.
The burning damage is updated every tick of the simulation and it considers attack speed, crit, multicrit etc.