What needs to happen is there needs to be more to the missions, so that they don’t end right away.
Maybe missions should be made in such a way to where they are sort of broken up into sub-missions throughout the week. What I mean is, for each sub-mission (which occurs in intervals throughout the week), you complete a specific task that leads to having something. Maybe it will lead to having one first form of the monster.
Then you wait for the next sub-mission, which will give the next fusion ingredient.
And the next… etc. So if it takes five monsters to fuse into the final one, you can have five sub-missions, each giving one.
Those sub-missions could either be dungeon runs, or some other type of activity (to add some variety).
This way, you’re more actively doing things throughout the week instead of it ending so fast.
How does this work with the whole pool idea? You can put a % restriction on each sub-mission, possibly. But that would take some deeper thinking.