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A Lego type approach would work for me.
Say you break the machine down into components such as;

Frame
Gantry
Linear motion (rails/bearings, rack & pinion, ball nuts, etc.)
Control Electronics...

I'd be surprised if you got that much a temp difference between screw and bed rail given they're only going to be a couple of 100mm apart or so.I'd turn the screw end down to say 15 or 12mm depending...

I used 50x50x3mm box section on which to mount my profile rails but first I looked up the dimensions of standard nuts. This gave a good guide to the depth of thread required for any size of bolt. So...

Ok I'll make this bit easier for you and tell you specs that I know work for this size and type machine.

Base frame: 50x50 or 60x60 x 3 or 4mm Steel. (Forget Ply wood waste of time)

Gantry:...