I know you are fearing the worst if you tell anyone about the problem, but it's not whining. You are far more effective and more of an asset to those you work with and work for if you can get through the day with less pain.
Try approaching it from that perspective. With the right chair you can do more and do a better job overall.
No one should be in such pain, hon. I hope the Temp Agency agrees.
Gentle hugs,
Alice