KmarkoPL loves to spread misinformation for some reason. Many of the weapons from the DLC are available for use in the main game and co-op. The shotgun in HoD is just a reskin of the M133 already available in the main game. That may be why they didn't bother to bring it over to the main game.
You can use weapons from DLC in co-op. It doesn't make it incompatible for the other player. They just don't get to purchase those weapons from the shops. You can drop the weapons for them to use, but I think they may lose it on Fast Travel.
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