Captain Zoom
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Without doing any rigorous research (I'm shooting from the hip), I'm gonna go with yes and no.
Yes, the Weapon Training itself stacks, but not the levels.
For example, if you have Weapon Training 1 from Sohei and get Weapon Training from Fighter, then you have Weapon Training 2 (+1 with one group and +2 with another).
On the other hand, for example, if you have 5 levels of Sohei, taking one level of Fighter won't get you Weapon Training as the levels don't stack.