A URL is the only way to go and apart from sheer convenience, it conveys that it's coming from Twitter, instead of the solitary "Tweet" at the end of the MLA's citation. Twitter isn't a timeless utility, it's a company running a service that happens to be popular at this time and in certain countries. You can't just say "Tweet" and expect every reader to always get the meaning of it.

I think it's okay to include the actual text too, as the short length makes it possible to do so and given that tweets don't have any title and are probably less likely to be preserved if Twitter went down than most other online references a scholarly article might cite.