Increase voltage output by connecting 2 PSU channels together

The culprit is not the LEDs themselves, they're harmless, but the TLC5940s, which switch at high frequency to control LED brightness through PWM. You can't filter the PWM outputs (you can, but then the brightness control doesn't work properly anymore), so that's out, but you can try to do something about decoupling power supplies. Not guaranteed to work, the fact that the scope's probe picks up the signal unconnected indicates that it's probably radiated, but it's worth trying.
Decouple the TLC5940s properly. They have to provide a lot of power, so that means 100\$\mu\$F, 1\$\mu\$F and 100nF all parallel on the power supply for each device, the smallest value closest to the pins.
Decouple your 5V power to the microcontroller properly: 100nF close to the pins.

For a variable lab bench supply you want one that has adjustable voltage that once set to a particular selection can supply a whole range of current to the load whilst the voltage stays constant.

Your first idea of using the pot between the +12 to -12V will not do do that. Pull any amount of current from the wiper of the pot and the voltage will change.

The second idea is a hack job modification done to the a switch mode power supply. From the description that you snipped this relates to a very specific model range of power supply and trying to apply this in general to any PC power supply from any vendor is not going to work. Not all power supplies use the same circuit configuration or even the same control chips. To be able to modify the behavior of any given switch mode power supply requires that you have good knowledge of how the circuitry of that supply is designed and how it works.

There is another reason not to be screwing with trying to make a switch mode power supply adjustable. Most switch mode controllers, especially the lower cost designs supporting high current outputs, are design optimized to vary the PWM of the switching output over a range to satisfy the varying current load range for the output. If you try to modify the feedback network to make the switcher produce other output voltages the design optimizations will no longer apply. As a result the supply may exhibit a number of undesirable behaviors including:

• The output may not adjust over a very wide range.
• The output may not be able to supply the full range of current at many output voltage settings.
• The switching regulator could become unstable and oscillate like crazy. This behavior could vary depending upon where the adjustment pot is set.
• The ripple of the output could be extremely high.

There could be other problems as well. Personally I think you would be better off purchasing a power supply specifically made for lab bench use. There are variable supplies available online that are fairly reasonable in cost. With these you also get the ability to vary the current that can be delivered to the load. This is a very useful thing when powering up a new circuit because you can set the supply up to protect the circuit in case there is an error in the circuit that would cause excess current flow that could potentially blow out some components.