The color space of the file has nothing to do with it. Any color profile should be displayed correctly - i.e. identically.
The BenQ software is known to be buggy and doesn't load the monitor profile correctly into the OS (where Lightroom should normally find it). I don't know if this bug has been corrected in later versions.
The video driver could also be buggy. If disabling it fixes the issue - keep it disabled until a driver update fixes it.
There can be several reasons Develop and Library show a difference. This is a conversion from the source color space into the same monitor color space - but the source color spaces are different: linear ProPhoto vs. Adobe RGB. That in itself is not a problem, as pointed out above - but it can mean that a botched conversion may affect them differently. If the conversion works as it should, they will display identically on screen.