I think it doesn't work with that, because these are not materials.
Not sure about relic dust ( regardless you get them from duplicated loot or just from chests ).

Mythic frags only drop on apocalypse difficulty as far as I'm aware. That doesn't mean you have to actually "play" through maps on that difficulty though. The game itself just needs to be set to that difficulty.

Meaning all you have to do is launch a campaign or adventure on apoc and Dwell will have a chance to give you mythic frags in exchange for relic dust assuming your power level is high enough.

If you want the best odds at mythic frags you also NEED to max out every single archetype. What quality of fragments Dwell sells are based on your character's power level and the only way to increase that after maxing your equipment is by leveling more archetypes.

I think only apoc drop mythic. Explorer can increase the quality of fragments that's true, but I think even the best nightmare fragment are no way close to mythic.