My story, and I realize its just one story but hopefully it's helpful. 39 years old and had been TTC my first for almost a year. Had a uterine polyp which wasn't likely causing any problems but I was going to have removed to be on the safe side. Got an HSG as part of an overall workup. Got pregnant the immediate first cycle after, no problems with the pregnancy and had a healthy boy. Starting trying again 6 months after he was born, because I knew I didn't have much time. Waited 5 months and then went in for an HSG, pregnant second cycle that ended in miscarriage. Six months after that asked for another HSG (Shady Grove) and got pregnant immediately after, healthy pregnancy and baby. If you google HSG and fertility there is some research (not enough) that an oil based HSG dye has a small positive correlation with pregnancy in the 3 cycles post procedure but no one seems to know exactly why. Basically, the only times I got pregnant were directly after HSGs (and I was using OPK, and an app to track my cycle) As for radiation, this article may help allay some concerns:
https://www.fertstert.org/article/S0015-0282(16)42194-1/pdf