Found a major flaw in paper from home university – to which I would like to return
Don't make it personal. Keep it about the science. You can inform them of the flaw and let them fix it or you could just publish a new paper with better results, pointing out the flaw if needed. That would depend on whether you want the new result under your name or are happy enough for it to be under theirs. 

If you write, however, and get pushback, evaluate it fully and then publish or not as you choose. But I wouldn't get into an argumentative back and forth about it. And if you just decide to publish it, as a courtesy you can send them your paper when you submit it to a publisher: For Your Information...

The science is the important thing.