As a simple practical matter, "hanakami" is incorrect Japanese. The "k" changes to "g" via https://en.wikipedia.org/wiki/Rendaku; there's already an example in "origami". So at the very least, it should be "hanagami".
(The fact that "it also means “tissue paper” in Japanese (鼻紙)" is irrelevant; homophones are common in Japanese.)
The choice of words is also kind of weird. "Oribana" would work, but that already refers to flowers made of origami. I would have gone with "hanaori" instead. "Hanagami" sounds like you're making paper out of flowers, or flower printed paper.
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