The idealist Paragon trusts her word that she will behave and avoid capture so he spares her.
The ruthless Renegade thinks that the risk is too high ("You're too dangerous") and kills her rather than take the chance of another Morinth or giving the Reapers getting another Banshee.
But the Asari government says she's dangerous. Being as we're operating within their space, we should follow their preferences, that is, swift elimination to contain the entity.
Sparing her is against protocol, something a renegade would do.
I mean, it doesn't make that much of difference, since Falere is just an Asari.
Where does it say that paragon is idealist? Paragons are by the book, renegades bend the rules as per the original attempted morality of ME1.
Take for example indoctrinated Cerberus Troopers. They are absolute paragons, following any and all orders relayed to them. Garrus is a renegade, taking matters into his own hands outside the boundaries of proper legal action.