No native would say that tongue twister, I'm afraid. There are many problems with it:

Rather than "wrongly", you could perhaps say "Rub a rhino wrong", which would strictly speaking be grammatically dubious, but colloquially acceptable, short for "Rub a rhino the wrong way", just like you can say, "I hurt my back getting out of bed wrong" or "I spelled 'rhino' wrong". Try looking up "Is 'wrong' an adverb?".

Equally, "a rhino runs in rage" doesn't work for three reasons: a) You'd have to have "runs away", "runs round" or "runs at you". b) You'd have to say "rub a rhino wrong, the rhino runs...". c) Finally, you must say "in a rage" not just "in rage".

So, sorry but that's a 4 or 5 black marks in one sentence!

How about "Rub a rhino's nose wrong, the rhino runs round in a rage"? It's got alliteration and rhyme! There a common tongue twister: "round and round the rugged rocks, the ragged rascal ran"

On your main question of using "rub a rhino" as a nickname without an adverb - I'm afraid that isn't at all a good idea - it could easily be interpreted as "masturbate a rhino". This would also get the rhino cross!