There are many, many pitfalls and challenges and a good company or individual can help you avoid them. A good partner can also help provide good ideas based on experience that will work. To directly answer the question, I don't know of anyway for a non-profit to mail one letter withou paying first class rates. There are pre-sort rates, etc. that still go first class, but you either need to sort them yourself or hire a company. For a small non-profit I don't think it would be worth it.
Assembly houses are as good as their last job.
Talk to a few of the charities in your area who have successful direct mail programs - and are about the same size. What a Muscular Dystrophy Association or American Cancer Society does for direct mail, probably doesn't compare to your anticipated program. Compare apples and apples. Some will not schedule you in until you physically deliver the piece - and then it will be another 4 to 5 days before it goes out. Some can schedule based on your estimate of delivery time.