Each ship has a cargo hold that you can dump stuff in. From the 2 ships I’ve been in (Frontier and Shieldbreaker) you can access the cargo hold from a computer to the left of the pilot seat. If you buy or find Ship Parts (the name of the item), you’d have to go to your characters Aid section, and move them to your ship’s cargo hold. Then the button shouldn’t be greyed out anymore.
At the moment (and in my experience as a fellow Necro) minions are just too weak to be much help in harder/higher content. There are only a small few builds I’ve seen where minions are viable, and they require very specific builds/items that would be hard to find at 40. Finding a build that sacrifices them is usually the better option. Although it’s been nerfed heavily, the bone spear build is still very strong and is what I used to solo the capstone at 42 this season. Not sure if there’s a difficulty difference running the capstone on wt1 or wt2, but if there is, my clear was wt2
If you didn’t have the carrying capacity, it would still duplicate and just leave it in his inventory.