No, the document profile and the monitor profile are two different animals, serving two different purposes. They should not be the same, under any circumstances, because each describes a different color space.
The sRGB profile in your document defines the colors in an absolute sense. Or said another way, the file is in the sRGB color space.
Your monitor has a different color space. To display correctly, the RGB numbers need to be translated into that color space. So you have a monitor profile that describes the monitor color space.
This translation is what color management does, by converting from one profile to another, thereby preserving the color appearance. Color management is about preserving color intent across different spaces and devices.
So you can see where this is headed: you need a monitor profile that accurately describes your particular monitor. If you calibrate and profile your monitor you will get that. If you don't, you will need to find the closest match. I don't know how close the "LCD Color" profile is, there's no way for me to know, but if it's the best you have use that. But let me stress this: if you want to see accurate color on screen, you need to get a calibrator to make a custom profile.
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Then the same goes for the print. I don't know how it's going to be printed, but the same principle applies: you (or the printer) need to have an accurate profile for the printing process.
If all these profiles are good, the print will match what you see on screen.
