Thanks, I didn't know I could force Windows to accept other color profiles rather than the screen's default. You put me on the right track and I could understand how to setup properly. If I set Designer to use the screen's color profile, the export would be the same colors than the profile's, not the colors I expected when loading up the palette in another software.
For future reference, here how I did it:
If Windows' Settings apps doesn't have this color profile in the Color Management panel, you can add it by going in Advanced display settings, then in the display adaptor properties link that'll pop a window. Click on the Color Management tab and the Color Management button. You should end up in the Color Management window with a list of color profiles set up for your selected screen. If the sRGB profile isn't in the list, you can tweak the default system settings by going in the Advanced tab, click on Change the system defaults and then click on the Add button. The profile should be on the list. You should be able to backtrack to the device's properties, use the sRGB profile, and have Designer display the proper colors.
Thanks for the help! Hope you have a nice day!