Does a Flametongue and Booming Blade work together?

I’m planning a magic initiate Rogue with the Booming Blade Cantrip. He will have a Flametongue as it is a higher level character. I already know that Booming Blade and Sneak Attack work together but what I’m unsure of is if the extra damage from the Flametongue would work as well. Example is at 5th level Booming Blade adds 1d8 Thunder damage to the normal attack and the Flametongue adds 2d6 Fire damage, so if I use a Rapier I would get 1d8 Piercing + 1d8 Thunder + 2d6 Fire + 3d6 Sneak attack (assuming I qualify for sneak attack) + Mod. I’m not concerned with the secondary damage from the target moving. This attack would average (with everything factored) 30.5 damage with a modifier of 4, Double with a crit. Would this work?