These days any immigration related matter is like winning in a black jack. The odds are pretty even. But the odds starts depleting when other stuffs start coming into picture.
In your case, it is the client letter. Now, does client letter guarantees you the visa? Certainly not. But it makes a very strong case since you are in a EVC model (is it EVC or EVVC?). Will they reject your visa or put you in a long wait with administrative processing? May be. The odds are against you at this point. Also, sometimes that email things works. But in your case, I would assume they will be extra careful since it is a federally funded client. I would strongly suggest you to 1) Amend your H1B to match the LCA and H1B, if you have not done yet. 2) Get a client letter from the client. 3) Worst case look for a new client who provides client letter and go from there.
My case, Same employer, same client since 2010. I had the client letter with me. They still gave me 221 and did a inquiry to my client manager and my visa got approved in 3 days. I was in canada last year. Going back again next month to the same location (I dont like to take chance in Nepal, my own personal , superstitious reason) with H1B extension with everything else same. Oh yah, I have similar type of client too.