colors of exported image change

Proof to Monitor RGB turns off color management in Photoshop.

In other words, you are most likely comparing to a photo viewer that doesn't support color management at all, and which will never display correctly.

To minimize the difference, you convert to sRGB. But there will always be a difference between color managed and non-color managed software. Photoshop is right, the other photo viewer wrong (assuming your monitor profile is healthy).

Color Management is not about matching color but controlling the change of color. What is the RGB color space? One RGB color space to another looks to the eye that the color shifts but color is remapping to another space. This creates shifts from the original pixel. It is better to see the color in the space it is created in then convert it to the final color space (CMYK). When you use Monitor RGB, you are viewing the color with the calibration profile of your monitor. sRGB is the standard RGB color space. It is best for having an image for print and devices with small shifts between the two. Adobe RGB is a bigger space designed to move to CMYK but has a great shifts between devices and print. The proof setup is just showing what it looks like when you convert to another color space. The RGB color space of your image is created in, is the most important.

Below are 3 images assigned to 3 different color spaces and converting one to CMYK from the original sRGB image.

Below are different color spaces