Is there a way to keep mission weapons?

Solution 1:

Unfortunately not, IIRC you are talking about the mission where you have to go to his shooting range. You cant keep those weapons but just dig into everything you can, you will find one pretty soon, you could also find other slag weapons like rifles and bazookas.

Solution 2:

You get that mission when you first get to Sanctuary, but elemental weapons are rare at that stage. I think it's deliberate - you get the elemental tutorial before you've come across too many elemental weapons.

Don't worry about giving up the slag weapon - there will be plenty more as you level up.

In a couple of missions you'll find a similar introduction to e-tech weapons.

If you really want to keep the gun bear in mind that it will be replaced by much better kit when you're a few levels higher, so you could just take it, use it, and then when you find a better slag gun turn in the Marcus mission.