There is a couple ways to look at this.
Like Tim said the firearm work will be cheapest for the 6mmAI, BUT the dies are much more expensive and making cases as well as Handloading is then mandatory.
With the 250 its a factory caliber. it is more expensive for the firearm work. But dies are cheap, factroy ammo is avalible. Also altho the Handi is plenty strong for either cartridge, the 250 is a lower pressure cartridge better suited to the weaker break open action of the Handi.
My choice would be the 250-3000 because thats what you limited the choices to. If it was me, I would chamber for a rimmed version of one of these. Something like whats made for the Contender pistol/carbine and be happy.
CW