No, a dwarf could not melee a Balrog. If you look at all the one-on-one battles between Balrogs and their opposing combatants, there is a consistent measure of whom those combatants are. In every case presented, they have come from Aman: Feanor, Fingon, Ecthelion, Glorfindel and Gandalf. When Gandalf states to the rest of the Fellowship "This foes is beyond you," I believe he is not referring to any melee skill per se, but rather a spiritual or innate magical power that is required in battling a Balrog; hence, only those who have lived in Aman have made the effort, and even then have died after defeating a Balrog.
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And your little sister's immaculate virginity wings away on the bony shoulders of a young horse named George who stole surreptitiously into her geography revision.
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