Is there anyone in your family who would be receptive? If so, talk to them first about whether they think it’s a good idea and how you might go about it. Or maybe they can do it, if they find this social stuff easier.
I have a vaguely similar situation with a nephew who’s 4 years old and shows signs of either mild autism or a language-based learning disability (SLI). I don’t know him well enough to tell – he repeats things people say sometimes when he appears not to understand them, but if that’s his only “repetitive or restricted” thing it may not be autism. My younger sister, who has struggled with anxiety issues and suspects she might be one of those missed mild cases of ADHD in smart people, has tried to bring it up with our highly defensive and distrustful older sister, to no avail. So I tried it, having been diagnosed with ADHD at 13 and my family was like, “finally, an explanation for it all” at the time. (They’ve also wondered later whether I could be on the spectrum as mild autism became better known in the noughties. I’m quite sure I’m not, but secondary social issues and mild sensory issues – in autism both of those, as I understand, are primary and at least moderate in severity – are fairly common in ADHD.) No luck: she insisted that my nephew was less “weird” than I was at that age and was going to grow out of it all, and if she sought a diagnosis too early, he might well not grow out of it. She even said my relatively late diagnosis was key to my being a functional adult, when I see it as the main reason I still think I’m just lazy, stupid, or crazy half the time!
I think what you should do, if your highest-confidence family members say there’s just no way to bring it up with the parents of this relative, is try to bring it up with the relative directly if you get a chance. If the kid is a preteen, they should be more able to understand than, say, a 4-year-old. And they might be more open-minded than adults who, even if they grew up around neurodiverse people, still think diagnosis is either bull or a last resort. In any case, it’s a good idea if the kid hears, at an age that’s still fairly impressionable, a message from a trusted adult that they are not lazy, stupid, or crazy, they have some minority brain wiring that they can learn to live with, but they might have to do things differently. If my nephew does not grow out of his issues, I hope he can trust me regarding that stuff, even though his issues are clearly different from what mine were at his age.