ASTM A 276 lists two types of strain hardened stainless bars-- Condition S for relatively light cold work and Condition B for relatively severe cold work. Type 316 and Type 316 N are available in either Condition. Properties vary with bar diameter, but yield strengths up to 100 ksi are possible with Condition B. You should discuss this further (including questions about machinability) with a metallurgist from Carpenter:
Strain hardened 316 is not necessarily a readily available material. Many distributors/mills will have some stock, however, it generally is a special mill run.
You can try to call Ugine Stainless and Alloys. We may have something in stock that may work. If not, we would be happy to quote.
Cold finished bar in 316 should be a slow ground ball. Go to Cartech.com. Distributors always discourage you from using the most correct material. They want you to sub-optimize over their inventory. Go to the manufacturers and get what you want. This will force the distributors to stock it.
