Wow, nice job so far. The press looks good and the attention to detail on the woodwork looks first class. I wish all first time posters provided as much detail at the get go.
Doubling the jack pressure is going to take you into a new realm of pain I suspect. Not insurmountable, but at a guess after watching the video, a lot of stuff is going to need to be made thicker and stronger. That will make a lot of it heavier and harder to handle.
Fortunately by it's clever design, the press has great availability for adjustment to accommodate an increase in height of the "load." At this point, just looking at the press, I am guessing that the beams (RSJ's) will handle the 20-tons, but it would pay to check the bending load using the calculators here on this site by plugging in the real figures.
The "plates" beneath the catch tray will need to be as strong as the "plates" at the top with the jack. Since you are isolating the foodstuff from the metalwork with what looks like Polyethylene or Polypropylene (plastic) sheets the choice of metal is not so critical. In that case, common old Mild Steel hot-rolled plate will do fine.
Once again, using the Calculators on this site, (start here... http://www.engineersedge.com/calculators.htm )you can determine the workable sizing for the metal. I would start with 1/2" thickness given that the cost difference between 3/8" and 1/2" steel plate will be minimal. You will need one plate at the top and another under the catch tray. The disk brake rotor is a good idea as it distributes the load quite well during the pressing.
On another issue, I am wondering why you do not use the screw portion of the jack when you first run out of hydraulic movement. Releasing the pressure then unscrewing the top and hydraulically press again. Screw the top back in, place another packing piece (disk brake etc) and go again. Every second "push" will be using just the screw portion rather than another packing piece.