Unfortunately, I don't know if there is a way to get ROM hacks on to a NES Classic (although I have heard there's probably a way.) Your best bet is to get an emulator and find this particular ROM on the ROMhacking website.

You can get ROM hacks on an NES classic if you use hackchi I believe it's called.

I actually have a few hacks I've made on my famicom mini.

@sullyrox i knew there'd be a way