The 2N2907 is a PNP transistor while IRF530P is a N-Channel FET. So the circuit should be different.

In the case of 2N2907 you source current to the LCD but with IRF you should sink current to the ground.

And of course the polarity of the control signal should be inverted for the IRF. Hope you are aware of that.

As for the noise, your circuit does not have any decoupling capacitors very close to your switching transistor (either 2N or IRF) and to the ground. Put a 100nF and a 10uF in parallel and I am sure things will be better.

Also ground loops etc will have massive effect on this. Do not trust breadbording for this. Noise will be much more on such circuits even with capacitors.

Ioannis