Champ is merely an older form of the word, in fact, that's basically the OED's definition, you have to look up chomp to get the meaning. I've never heard it used outside of the idiom. I think it makes more sense to use the idiom with the newer form, to use words that are actually used.
JeffDM, I have before me the New Webster, Oxford, Britannica, Canadian, Larrouse, and a couple more dictionaries. Only the Canadian has the word 'chomp' in it. The others show , 'champ' to bite and chew noisily, and list a second definition, "to show impatience."
In the Canadian dictionary, "Chomp' is exampled by, 'chomping on a cigar' as in chewing repeatedly, whereas 'champ' is exampled by, 'champ' at the bit' as in showing impatience at being held back.
As such, chomp doesn't appear to be an old form of champ.