Dany won’t burn Aegon because: she is not a pyromaniac, has no proof that Aegon is fake, Aegon has done nothing to piss her off (even killed/defeated her enemies for her), Aegon is a kinsman offering her an alliance (burning him is not exactly great PR), Aegon is a Targaryen with public support etc. I can see potential conflict but it goes against Danny to do this sort of thing

Totally disagree. 

Dany’s decisions about a lot of things are driven by prophecy, as we see with the “three treasons.” And in Aegon’s case, Dany has prophetic reasons to believe that Aegon is false – the mummer’s dragon – especially when she turns up and Varys the Mummer shows up by his side and realizes that Varys and Illyrio have been double-dealing with her. 

And in terms of pissing her off – he’ll have taken her throne. Yes, he’ll be offering her an alliance, but as one of his (multiple) queens. Dany is not about that any more. She’s the Mother of Dragons, the hero heralded by prophecy, and she is done making peace.