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The source of the light is not even (since it's an array of LEDs) , but I want that 6x6 inch destination to be a smooth light.
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I can't access your youtube link (due to employer's firewall), so I haven't seen your experiments.
One idea I can suggest, for smoothing the intensity variations of the composite beam, is to mix the individual LED outputs using a small integrating sphere, and use a small output port of the sphere to illuminate the target, possibly via a lens.
The sphere could be small, cheap, and DIY, internally coated with white paint (TiO2) from an Arts supply store. The LED outputs could illuminate the sphere via cheap optical guides e.g. plastic optical fibres.
DIY integrating spheres are interesting per se, and may help you with your experiments.