There may be multiple problems. One is the colour profile PL is using. In Monitor profile it was asked what profile PL is using where there are more than one monitor. No one has given a reply so that may be a problem.
Different results in the prints can also come from how a program is using the profiles, I don’t know if you would get the same if via the printer. I have seen slightly different results between Affinity and PL for example.
Managing via the printer is an option in the PL print menu.
The only way I have been able to get prints that were OK was a rather complex work round. I export to Affinity, use its Adjustments. Soft proof with Gamut Check selected and the paper profile. This shows me where there are problems, usually there will be some clipping and you can see what needs changing in PL. Have PL on a second screen and adjust mainly with the tone curve using a virtual copy and export that and try again. In effect I am using Affinity for most of the print checking. In PL I find it spaces the side margins wrongly so I have to add 4mm on the left and take off 2mm on the right. This way does leave me with the option of copying the finished results to the master thus the dop. Another was would be a renamed copy of the image with the printed changes applied to it.
It does all raise the problem of the lack of proper printing PL and the problem of sometimes needing more than one “right” result not be coped with permanently. I know some export resized Tiffs for different print sizes and uses but I just would find that a real pain as taking up one hell of a lot of HD space. It would be much better to be able to save different version processes but PL doesn’t cope with that and I don’t know if any one does.