I think you'll need a 4-axis machine if you don't want to go down the laser route as you'll need to turn the blade in the direction of the cut (it might be possible to have a freely rotating swivel blade, but I'm not sure how well that would work). As mentioned, you'll have a limited cutting radius - but then you've probably already got that with your manual cutting.

It might be possible to use a mill-like cutter, but it could get very messy, and the edges may not be as sharp as you'd like.