Everything but eruption, will-o-wisp and heat wave should work fine.
Eruption and Blast Burn work especially well.

Damn it. Lava Plume doesn't really work but eruption might, if he can't stand the spiceyness of the food he's eating? Or something... other fire moves are ember, fire blast, overheat, will-o-wisp, heat wave and blast burn.

flame thrower yes
hyper beam no
try some other fire attack that doesn't involve punches or wheels

Just a question, with your challenge, a cyndaquil, that must know smokescreen, toxic and two other attacks that 'reflect it's heartburn', would flamethrower and hyper beam fit that category? If not, could you please recommend other moves that would?
Thanks.