The flamethrower is very powerful in the hands of a high-level FB, same as the xbow, and is pretty cheap with the perk discount so it should weigh a lot for balance. The solution isn't making it lighter, but giving the FB a lightweight secondary weapon that he can use with the flamethrower, providing flame coverage in a limited fashion (the backup should not be comparable to the flamethrower).
The Handcannon would be a fine candidate for incendiary ammo, but NOT a gun with a fast rate of fire, because that would be too similar in effect to the flamethrower, in that it would provide rapid area fire coverage. The Handcannon would provide short bursts of fire coverage and single-target fire coverage from range, particulary useful for burning high-value targets when flamethrower is spent and also lighting single targets when there are no groups around, something the flamethrower is NOT good at doing.
The FB would use the Handcannon for single targets and bust out the flamethrower for groups. This would bring much-needed variety to the perk and extend flamethrower ammo, keeping the FB effective in later waves without having to stockpile multiple flamethrowers.