In addition to the Hornady tool, pick up a 5/16-36 tap and a 7.3mm drill bit so you can make your own brass that fits the tool, since they don't make every cartridge for it.
Then get yourself some cleaning rods, or a whole cleaning set. They tap the bullet out fine if it gets stuck.
Rather than a new barrel, a gunsmith can probably cut and re-chamber that barrel, or if it's on the other end, just cut and crown, or ideally go ahead and thread it. You'll loose a few inches, but probably money ahead. Especially on a patriot.