Based on some reasoning, I've got this list:
Vanguard = Kinetic/Laser, as you said
Dvar = Kinetic/Explosive, also as you said
Kir'Ko = Psionic/Biochemical, ^
Amazon = Laser/Biochemical, putting in Biochemical due to their genetics knowledge
Assembly = Arc/Kinetic, Kinetic as you said, and I'm presuming some race needs to be Arc, and a race of machines could work, even if they might also have a weakness to it
Syndicate = Psionic/Arc, two psionic races, so that's why they have that, and it gives a second place for Arc in the lineup
One Explosive, since you mentioned only Dvar had it as a primary, and three Kinetic based on the provided list, making sure the rest of them were represented twice. Seems like a reasonable guess of the setups.
So, how linked are the terms "damage channel" and "weapon tech," since as we've seen a damage type of 'Thermal,' but no direct weapon tech? Several weapons which would obviously fall under the category of Laser have been listed as dealing Thermal damage, but the Phoenix Walker most certainly isn't utilizing lasers to deal its Thermal type damage.
Likewise, is there still a chance for what I suppose might be called "Cryo" damage to show up? Most of the weapon techs have fairly direct analogs:
Kinetic = Physical
Psionic = Spirit, sort of
Biochemical = Blight(acid/poison), but can technically induce effects similar to fire and shock
Laser = Fire
Arc = Shock
Explosive = ...Fire/Physical?