A brand new machine build. 2x4 size or something in that region. :)

[Jonathan]

Since the machine is no more than 1200mm wide, then the stiffness of a UNP 160 beam should be more than enough.

Have you calculated this to know for certain? If not, then this spreadsheet may be of assistance. You just need to know the length of the Z-axis, to work out the torque applied to the beam and the force which can be approximated from knowing what materials you're intending to cut. Also, even if it does turn out that the UNP 160 beam is strong 'enough' that doesn't mean you should definitely use it since the formula's for torsional stiffness make it obvious that using a closed section will require a lower volume of material than a C shape.

If I were to change to RHS tubes instead, then I'm gonne sacrifice something else..

RHS is just one idea, I'm sure you can come up with others with good torsional stiffness.

So if you think that you can bend a 1200mm long 160mm UNP beam, over a max length of 1000mm, then go ahead, I believe that deflection will be set at a minimum.

No, I don't believe I can bend it - it's maths not belief, it's twisting not bending and it's not a question of if it will bend, it's a question of how much. With a CNC-Router the structure needs to be very stiff since although the magnitude of the forces is quite small, even a very small deflection can cause problems (poor finish, tool wear, limited MRR), unlike most applications where we're worried about the material failing which is not a problem here.