You're basically right. The kami in origami means paper 紙 while the kami in kamikaze means god 神. That doesn't mean you couldn't make a pun about it, but the more fundamental problem is that kamikaze does not mean "suicide attack" in Japanese. The actual word for it is tokkou 特攻, a euphemism that means "special attack".
Kamikaze originally refers to this historical event where a "divine wind" protected Japan from invasion. The term was later used in relation to suicide attacks, but I think tokkou is the word that's more commonly associated with the WW2 suicide attacks in Japanese and kamikaze has kept its original definition.