I have been diagnosed with celiac for almost 10 years and I had elevated blood protein when diagnosed. I had other wonky blood work also, like elevated liver enzymes. All resolved on a strict gluten-free diet.
Celiac makes the liver angry and it is very common to have elevated enzymes for awhile after diagnosis. It can take quite awhile for these things to normalize. Doctors are just going to run other blood work to make sure it is nothing more serious, which can happen with long undiagnosed Celiac. You were eating food that was totally detrimental to your health so "stressing" out your internal organs happens. I was 46 when diagnosed and went a LIFETIME with symptoms but I am fine today and I am no spring chicken, either!
The blood protein issue was the same with me. I had elevated protein, not bad, but out of range of normal. So, off to the hematologist I went. I did a lot of research on it beforehand and had a good conversation with the doc about it. I actually found research which stated that Celiac Disease can raise protein levels and the electrophoresis test is important because elevated protein levels can signal Multiple Myeloma. I also have 3 other AI diseases from going so long without a diagnosis so it was pretty much a foregone conclusion my blood work would be wonky. (I really love that word!) It took quite a while, around 4 years from diagnosis, for that to normalize but I think it was the combo of 4 AI diseases which made it that way. I am an optimist to the core who hates going to any doctor for anything so I did not let it phase me after doing the research. I felt good at the time so didn't worry unnecessarily about any nasties happening.
The immune system is made up of proteins so when this goes haywire and your immune system starts to attack various body parts, it is more or less hyperactive. It would make sense that it could raise overall protein levels due to all that inflammation. As long as you don't cheat and eat a healthy gluten-free diet, the inflammation calms down over time and levels normalize.
I still have some wonky results like low white cell counts and my red cell counts will never be stellar (low end of normal) but I feel good, exercise and eat right so don't worry about it. Low white cell counts are normal for people with AI disease and it is only a problem if a person is getting sick all the time.....which I am not.
I do not know how long you have been diagnosed or on a gluten-free diet but have patience. If your levels normalized, it is highly likely they will stay that way...if you follow the diet as you should. The reason you had more testing is because you went to a GI doc and they tend to be more thorough, but that is not always true. I never went back to a doctor, except my thyroid doc, for any testing because I did not have a PCP at the time, just my thyroid doc. I had her run the Celiac antibody re-check yearly. It's a long story but I got screwed over like many people did over the years and still have trust issues with the medical profession. I found a great PCP about 3 years after diagnosis and she ran a slew of tests and found the elevated protein. I think it all depends on a persons experience in diagnosis on why some have more testing done than others. I personally just do testing that I think is important because I am not spending the rest of my life running off to doctors. Now that I am healthy, I have a life to live!