There is no such thing as “conflicting damage types” in DnD. Never has, never will. This is not Final Fantasy or Divinity Original Sin or Genshin Impact. Actually even Final Fantasy doesn’t have conflicting damage types.
If you want to house rule it you’ll need to make up your own rules for how it works. DMs are more than empowered to do so. The DM can make all the calls on how it works.
For the record, in real life if you freeze someone then burn them they still die. Once while cooking some hot oil got splashed on my hand. It hurt like a bitch I quickly stuck my hand into a basin of cold water. Guess what? The pain didn’t go away. If I instead stuck it in the freezer? Same shit. The burn is already there. No amount of ice will fix that. And if you keep it exposed to really cold temps it’ll get injured further. That’s just how it is.
“Conflicting damage types” is not a thing anywhere except the most esoteric video games. And DnD is not one of them. At least, not out of the box.
Your DM (or you, if you are the DM) can make conflicting damage types a thing. That’s something the DM will have to decide.