Using the Save as Render checkbox will apply the colour transform to the saved image, so it’s more likely to match the render window when viewed in Gimp.
Blender assumes sRGB colour but doesn’t assign a colour profile to the saved render, so to ensure the saved render is correctly interpreted, the first thing to do in Gimp is Image - Colour Management - Assign Color Profile. Use a reputable sRGB.
Your printer might want images supplied in a different colour space(AdobeRGB eg). If so convert to another colour space after first assigning sRGB.
The only sure way to display data the same on different hardware is by profiling, using a calibration device. Some cost more than others, but you may find the cheaper versions from the same brand are hobbled by the software supplied. The more expensive version may use the same hardware, and the software allows you to access more features. You may be able to access more of the features of a cheaper version using DisplayCAL and argyll instead of the supplied software - https://displaycal.net/
The calibration would have to be run separately for Windows and Mint.
Once you have a monitor profile you can use it in imaging software so that all your applications display data the same way. It should also help to ensure the image looks the same on the printer’s monitor as it does on yours.
Troy Sobotka has made available information on how to enable the monitor profile for Blender. This was written before the ‘Filmic’ LUTs were added to the Blender trunk, but it is still very helpful - https://wiki.blender.org/index.php/User:Sobotka/Color_Management/Calibration_and_Profiling
Using the monitor profile with the ‘Filmic’ LUTs is more ‘involved’.
If you want to find out more there is a wealth of information at Blender Stack Exchange, and down the rabbit hole.
Once all that is set up, you only need the print profile for soft proofing, and to figure out a way to squeeze the data displayed on the monitor down to the even smaller print scale.
You could just eyeball it, and ask the printer to make an adjustment once you get there.
Hope this helps.