While I'm not certain for the Store TP weapons (though I'm fairly certain that the Store TP from the gun would affect your normal melee swings), the TP bonus ones only affect weaponskills belonging to those weapons. As such, the gun would only affect Marksmanship weaponskills, and wouldn't boost the effects of any weaponskill you tried using with your mainhand weapon.
Lady Jinte wrote:
Vlorsutes' Negotiation Skill rises 0.2 points
Vlorsutes' Observant Parent Skill rises 0.3 points
Vlorsutes' Argument Diffusing Skill rises 0.1 points