It's the fire that ignites objects, not the damage type
Maybe by strict rules as written, the feature only changes the damage type, but this outcome makes zero narrative sense. How do you explain in-universe what is happening here?
Fire damage (p. 196, PHB) is described as
Fire. Red dragons breathe fire, and many spells conjure flames to deal fire damage.
If you change the damage type and do not deal Fire damage, you are not conjuring any flames to do so.
Does the bludgeoning damage now dance upon the object and damage it? Does the psychic damage consume it slowly? Even with good will and creativity, it is hard to see how that could make sense.
This is a situation where the DM should exercise their right to overrule a nonsensical result of the rules, as advised by the DMG, p. 4:
As a storyteller, the DM helps the other players visualize what's happening around them, improvising when the adventurers do something or go somewhere unexpected. As an actor, the DM plays the roles of the monsters and supporting characters, breathing life into them. And as a referee, the DM interprets the rules and decides when to abide by them and when to change them.