Subjugating bandits to gain rep? Sure.
Letting them go to gain rep? Fuck no.
These are, by practice, some of the most motivated men in your entire country. They are willing to become outlaws and engage in violence to make a living...fucking conscript them to your micro army. It is a waste to let them return to just being peasantry when they are so very obvious willing to fight to stake their own claim in this world.
You have internal enemies just as you do external enemies, and they can't simply be beaten by showing the people what a nice guy you are. They are your competition and mean to see you dead no matter what so long as you retain any semblance of power, so you need to gain more power.
Alexis is a luxury, for you, it's fine to dream about retaking it one day. First, however, comes the purging of your traitorous allies. Unless, for some reason, you can magically beat the foreign army on your own-Alexis comes last.