A friend of mine works for a higher instituion which like Non-Profit Organization is not affected by the H1 quota. According to him, the rejection chances are also very low. I think he paid for his H1b fees. It was $1000 for premium process (less than 2 weeks). I am not sure about the lawyer part but in his work place there was someone who did the paper work for him. So in your case, if your institution has sponsored other foreign nationals before, they will know who and how to do the paper work. But yes, don't worry about the quota part.