And if you don't like the other weapons that the ADVENT Flamethrower adds, you can use these lines:
+WEAPONS_TO_HIDE=ADVENTChosenPistolConventional
+WEAPONS_TO_HIDE=ADVENTChosenPistolMagnetic
+WEAPONS_TO_HIDE=ADVENTChosenPistolBeam
+WEAPONS_TO_HIDE=ADVENTChosenRifleConventional
+WEAPONS_TO_HIDE=ADVENTChosenRifleMagnetic
+WEAPONS_TO_HIDE=ADVENTChosenRifleBeam
+WEAPONS_TO_HIDE=ADVENTChosenShotgunConventional
+WEAPONS_TO_HIDE=ADVENTChosenShotgunMagnetic
+WEAPONS_TO_HIDE=ADVENTChosenShotgunBeam
+WEAPONS_TO_HIDE=ADVENTChosenSniperRifleConventional
+WEAPONS_TO_HIDE=ADVENTChosenSniperRifleMagnetic
+WEAPONS_TO_HIDE=ADVENTChosenSniperRifleBeam
+WEAPONS_TO_HIDE=ADVENTPriestRifleConventional
+WEAPONS_TO_HIDE=ADVENTPriestRifleMagnetic
+WEAPONS_TO_HIDE=ADVENTPriestRifleBeam

+WEAPON_REPLACEMENT=(ACCEPTOR_TEMPLATE=ADVENTFlamethrowerBeam, DONOR_TEMPLATE=FlamerMod_CV, COPY_DONOR_ATTACHMENTS=True, HIDE_ACCEPTOR_ATTACHMENTS=True, HIDE_DONOR_TEMPLATE=True, HIDE_DONOR_SCHEMATIC=True)
+ANIMSET_REPLACEMENT=(ACCEPTOR_TEMPLATE=ADVENTFlamethrowerBeam, DONOR_TEMPLATE=AdvPurifierFlamethrowerMP, REPLACE_WEAPON_ANIMSETS=true, REPLACE_SOLDIER_ANIMSETS=True)

+WEAPON_REPLACEMENT=(ACCEPTOR_TEMPLATE=ADVENTFlamethrowerMagnetic, DONOR_TEMPLATE=FlamerMod_CV, COPY_DONOR_ATTACHMENTS=True, HIDE_ACCEPTOR_ATTACHMENTS=True, HIDE_DONOR_TEMPLATE=True, HIDE_DONOR_SCHEMATIC=True)
+ANIMSET_REPLACEMENT=(ACCEPTOR_TEMPLATE=ADVENTFlamethrowerMagnetic, DONOR_TEMPLATE=AdvPurifierFlamethrowerMP, REPLACE_WEAPON_ANIMSETS=true, REPLACE_SOLDIER_ANIMSETS=True)

+WEAPON_REPLACEMENT=(ACCEPTOR_TEMPLATE=ADVENTFlamethrowerConventional, DONOR_TEMPLATE=FlamerMod_CV, COPY_DONOR_ATTACHMENTS=True, HIDE_ACCEPTOR_ATTACHMENTS=True, HIDE_DONOR_TEMPLATE=True, HIDE_DONOR_SCHEMATIC=True)
+ANIMSET_REPLACEMENT=(ACCEPTOR_TEMPLATE=ADVENTFlamethrowerConventional, DONOR_TEMPLATE=AdvPurifierFlamethrowerMP, REPLACE_WEAPON_ANIMSETS=true, REPLACE_SOLDIER_ANIMSETS=True)

My WSR Config has the following lines (posted in multiple comments for the sake of beating that damn 1000 character limit):

Someone in the comments of Flamer 2 has you covered, he has all the stuff you need to use it with WSR and ADVENT Flamethrower.

Damn it wanted the flamer thrower but the Flamer 2 mod doesn't work with WOTC.