Strictly speaking, 폭파 means "destroy by explosion" (폭: explode, 파: destroy), although the two are sometimes used interchangeably.
시키다 is "make something happen; trigger something" - it means bringing another entity to an action or event (교육시키다, 파멸시키다, 진정시키다).
So 폭파시키겠어 means "I'll make it explode / I will explode it". It is a transitive verb so you need an object for it.
If you eat too much and feel like your stomach might explode, (나) 폭발할 것 같아 / 폭발하겠어 is a better expression.
But even that is not the best way to say it. We say 배가 터질 것 같아 / 배(가) 터지겠어 (My stomach might explode) after having a big meal.
터지다 and 폭발하다 have the same meaning, but 터지다 is much more common in everyday life because it is a native Korean word, while 폭발하다 (hanja-based) is used more for real and serious explosions.
Anyway, your playful guess was quite good except for the word choice.
"배(가) 터지겠어" is quite often heard in real life :-)