I can't relate, since I don't have anything serious at all with my health, however, I can say with certainty that you need a diagnostic biopsy to rule this in or out.
Ref: IgG4-related ophthalmic disease
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4314568/
And because you already have a diagnosis of Hyperthyroidism, your doctors are pointing to Graves’ Ophthalmopathy (much more prevalent than IgG4 ROD) , where the overactive thyroid affects the area surrounding eyes and may cause inflammation, redness, achiness, pink eye, eyelid retraction and bulging eyes, etc.
Best wishes,
Niko