To defeat Brother 6 with the flamethrower, start by either running a circle around him until he stops shooting fire or hit you with the fire. If he hits you, he will laugh and be open for an attack. Use Slash to stun him, then use the SQUARE(4) combo until he shoves you away. After hitting the ground, immediately use the Slash. If done quick enough, you will interrupt his attack and stun him. Repeat the combo on him, then repeat the entire process through all of the phases.
Time your blocks to parry his attacks and leave him open. You can foresee a big attack when his weapon starts sparkling. Use any combo until he begins blocking. After you have locked swords with him twice, he becomes faster and stronger. Keep parrying him and combo back if you are skilled at timing your moves. Otherwise, Focus Dodge and attack.