If you're using the indefinite article, then surely you'll need a noun, so he would be a Lebanese EnglishMAN? As for word order, I think there are transatlantic differences. While I can't speak for the UK's Lebanese population specifically, I'd say that while US usage would put "American" last, the UK and its constituent parts would often put the more recent nationality first. Not always, but often. In this particular case, both versions sound natural to me.