I think it's going to be quite a challenge to get a dedicated free standing cabinet that small. As you said, i would go with the wall mounting one. Usually they are all sturdy enough, but i would go with the rittal, as they have a bit thicker material and superb build quality vs others.
Those server cabinets (IT cabinets) usually don't have a mounting plate as an option, so you would have to custom make one and also figure out how to even mount it into the cabinet...
I would recommend Rittal AX 1360.000, which is 600x600x350. You mentioned "at least 350mm depth". Keep in mind that the useful depth according to datasheet is 328.7, measured from mounting plate to the door.
As it will be placed on the floor, it will be quite difficult to get the cables in there through the cable glands. I would order gland plate SZ 2567.400, which has foam rubber strip for the cables and clamp strip to secure them down.
For the plinth you will need to make some sort of custom solution, as there is no accessory option for this cabinet. If you could live without the side panels, you could try with AX 2820.061, which is plinth for 600mm wide cabinets. Of course you would need to drill the holes in the bottom of the cabinet to mount it. Check the CADs of the parts how it would fit together.
I don't know where the cabinets will be mounted and how much weight will it be on the mounting plate, but consider about securing it down to floor, as it is relatively shallow and it can quite easily tip over, especially if there is not a lot of weight on the mounting plate. I even think that the cabinet of such dimension will tip over when empty, if you open the door...