I check my oxygen level with my home oximeter when I am on the sofa and everything is fine. I took my oximeter with me for my last test and it showed the same results as PFT. But home oximeters are not as good as the ones they use in pulmonary testing so using them when moving is a little more iffy. I use mine when on a treadmill and I am able to keep my hand still on the bar. I do go down to 88 (even lower if I increase the speed).
I had a cardiac stent and did cardio rehab. They had sturdy professional oximeters, and they showed my oxygen dropping while on the treadmill. And my pulmonary tests confirm that my oxygen levels drop with exertion. The issue is why oxygen level drops as it can be lungs, heart or both.
Ask your doctor to order oxygen titration test to have it documented.