Update on the design. I have finally managed to get sufficient stiffness in the Z cart. The solution was no surprise, maximizing second moment of area, and... steel. I also moved the Z blocks down as far as I could and placed all the Z blocks over the Y blocks. I am now planning to use a 100x200mm t=4mm square steel tube. The Z cart became twice as heavy, now 20 kg (44 lbs). However, about 20 times stiffer. The large cut on the backside is so I can get more space for the ball-screw and eventually remove the spacers for the Z blocks. I did test if the cut would compromise the design but it held up well to loads in all directions and torsional load.
Click image for larger version.  Name: Ny gantry.PNG  Views: 553  Size: 62.1 KB  ID: 18373

Now that the Z cart is good, torsional stiffness in the gantry is the problem. I am also doing the analysis on the whole gantry with Z and Y carts. The connections between the blocks and rails is not correct. They are just bonded. I do not know of any good ways in SolidWorks to make them in an efficient way, without messing around with spring connections. Therefore, it is a gross simplification, but it should give an approximate overview of the deformation.
Click image for larger version.  Name: Gantry med Z akse 100N X.PNG  Views: 993  Size: 91.3 KB  ID: 18374
The image is with both gravity and a 100 N (22.4 lbs) load in the X direction, a potential worst-case load. It gives a deflection of 0.014 mm (0.55 thou). With some vibrations, joints, and the base, the deflection will be bigger. At this point, the structural frame is not the weak link any more, but ball-screws, rails, steppers, drivers, and so on. The stiffness is at 13N/um now. As I am a newbie on this, I wondered if this is a good point to settle?

The gantry, Z and Y carts weight about 40 kg (37.8 lbs) with a aluminium gantry. And 65 kg (143 lbs) with a steel gantry. So I guess steel gantry is not out of the question.

I also looked into changing the design on the gantry, by using two parallel beams and having the Y and Z cart mounted in the middle. Like Routercnc's MK4 router design. This had many advantages on the rigidity and weight distribution.
Click image for larger version.  Name: Paralell skinne gantry 100N dobbel X.PNG  Views: 756  Size: 88.1 KB  ID: 18375
However, this would require rails on both sides of the Z cart and I cannot find any good way to prep the surface and mount them accurately. With the classic gantry design, I can use epoxy, but with the parallel beam design, the Y and Z cart becomes a problem. The design would also be a lot more complex with parallel ball-screws on Y. In addition, four rails on Z or parallel ball-screws here too. And I have enough problems to worry about already, and on a first build it's best to keep it simple.