I would suggest that you get another opinion from another spine surgeon.
Why are you seeing a rheumatologist?
It may very well be that your doctor is counting the vertebrae in your spine differently as there is no uniform method of counting the vertebrae and numbering them.
It could be that you are not missing the 5th vertebrae but rather have, what is called, a transitional lumbosacral junction. that is when the last vertebrae area is fused to the sacrum and if not recognized by the doctor or the radiologist, they either give a different number to the vertebrae as they appear or just assume that you are missing a vertebrae.
this is something that happens in utero during fetal development and is not that abnormal to have. It just makes a problem in counting the vertebrae as there is no uniform method of counting.
You should get another opinion from a sports medicine orthopedic surgeon who specializes in spines.
And you should be referred by the sports medicine ortho to a sports medicine physical therapist that does heat and stim to relax the muscles, then ultrasound message to the are of the joint that is a problem for you. And also does myofascial trigger point treatments.
then add a very slow strengthening program for you to strengthen core and and quad muscles.
I am surprised that you have not been offered steroid injections under floroscopy into the S1 joint to relieve some of your pain.
It may be that some, if not all of the pain you are feeling is from inflammation if the celebrex is working to relieve some of the pain, but there may be more going on and that is why I suggest that you seek out another opinion with a different spine surgeon.
It doesn't sound like you have been getting the best advise or medical care from the present doctor you are using. Telling you that it is only mechanical and will go away on it's own, when you are in the pain you are in just doesn't sound right.
Good luck and let us know how you are doing.
Fran