Yes make the gantry 1600 and things get much much easier.

I think for a router you will still possibly need 2:1 reduction on the drives though if using rack/pinion, ball screws would be ideal but much more money.

Triangulation - draw a long thin box - this is your beam, now draw a triangle on top spanning end to end of the beam, from the apex draw a vertical line down to the beam. You now have a box with much more vertical rigidity, triangles are very strong when used right. The trouble is this will not stop twist - you will not stop twist with a single beam that long - it will need two parallel beams with the spindle/z axis mounted between them, this will be very sturdy but 2 x heavier - there is no free lunch here :)

I doubt you will achieve 1mm tolerance in the middle of a 3m beam, plus you have lift - when the cutter plunges into the wood, the beam will lift causing tapered cuts etc.

Big machines need lots of design i'm afraid, and lots of money to make them work.