Splitting voltage and current unevenly
You need to clarify what you mean by "independent power supply"

If you mean the LCD needs an appropriate voltage to run, and that voltage must be fixed, and independent from the voltage output by your power supply, then yes - you can just use your laptop adapter to generate two separate output voltage lines. One of them with low current (assuming LCD draws milliamps) and fixed voltage. The other a variable voltage, where the laptop's 19V are reduced using either a linear regulator (simple but difficult to construct for larger amperage), or a buck regulator - much more difficult to implement but can handle larger current without an excessively big heatsink and fan.

If you mean the LCD needs to be galvanically isolated from the voltage provided by the lab supply - it is going to be very difficult. You need to convert DC to AC, pass it "through" a transformer, get it back to DC. This is a tall order for anyone who doesn't specialize in PSUs.

Maybe you are actually asking on how to measure the voltage / current of your supply, and display those on an LCD. You should improve your question by adding a circuit you are considering for the actual "supply" part, and then we could help about the LCD portion.