It is going to be much more significant how the floortray is attached than whether it is thick or not.
Think of cardboard box. It cannot hold its shape very well until the bottom is in place and 'locked'. Even a weak bit of box tape across the middle of the bottom flaps has a disproportionate effect on the stiffness.
A floortray isn't going to have a very significant effect on vertical distortion unless you are comparing two significantly differently stiff materials. Sit a 1 kg block on the end of a floortray supported at one end and see how much it bends. Compare them.
But when one side of the kart moves independently of the other it tends to 'lozenge' the 'box' containing the floortray, and a floortray that cannot move relative to the box will resist that more than a floortray that is lightly secured.
I suspect you will only see a significant difference if your current floortray has holes that are distorting under the stress.
But I may be wrong.
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