OK, got it.
Can you add an earth ground (AC power socket ground) to the PSU negative output?
The switch electronics need a good enough ground (literal or capacitive) so a decent level sense signal is applied to the sense electrode, relative to that ground.
Most things use the bulk electronics as the ground with a relatively small touch contact; your setup is almost inverted from that! It needs some form of reference ground, to be able to both drive the sense signal and compare the touch contact signal to the electronics ground.
Do you have an oscilloscope, so you can look at the signal and the noise level & frequency on the casing & PSU output?
Other than that, I think you need to try different PSUs to find a type that has a cleaner output with no common mode noise.