Colorspace coversion is what it says, convert from one to one, and needs a rendering intent. When yo assign a colorspace, thus not convert to, then you get washed out colors if you do that for an image that should have profotoRGB assigned, but gets sRGB assigned. vice versa if you assingning pRGB or even aRGB to a image that should have sRGB assigned colors look overly saturated.
I knwo little of mirage, but softproofing in LR is to see where the printer will short-change you in colors, thus allow you to change the image such to compensate for that. This changed image can then be exported to tiff in pRGB colorspace and this exported image can be printed with mirage with the printprofile as you want it.