After about 30 minutes of testing I can say that Infusion dose not work properly with The Companion legendary Jakobs pistol. As you all may or may not know Infusion dose have a chance to proc elemental damage over time effects that do damage proportionate to the Infusion action skill element damage. With The Companion the gun has a zero percent chance to proc elemental DOT effects but auto ignites enemies when you land a critical hit. Nether a Crit or a body shot can cause any DOTs from Infusion.
I am not sure if this is a bug or intended but it could have been a really cool combo.