You want to be feeding Specters the strongest guns in the game. So that's going to be
Crow-Sourced.
For non DLC options you can try
Swan Song,
Stray,
Flash Cyclone, etc.
Ghasterbait for Heavy weapons are an option. Try
Draupner or
Heimdahl.
MIXOLOGIST TECH:
After summoning a Specter, swap to a Beam Tosser underbarrel on Maliwan Shotgun while using a Mixologist Maliwan enhancement for massive Specter damage, at the cost of not contributing yourself.
Rekpit:
Triple Bypass with Accelerator for
Now With Caffeine and cooldown.
Grenade: Jakobs throwing knife with Penetrator augment for crit chance.
Enhancement: Powershot for your Crow Sourced. Or Mixologist for the tech above.
Shield:
Little Sister provided the most, can run Boxer augment then a source of melee damage or enemy damage to trigger it.
Firewerks for general mobbing.
Class Mod:
Living Weapon for the extra kill skill stacks from
Grave Power points provides for both you and your Specters.
Spirits with
Grave Current for maximum Specter damage. The bonus element goes up into the 100s.
Illusionist Budget use until you can get
Living Weapon online.