Yes, he would be focused on his instruments, anyway, but he should have diverted or, if not possible, should have scrubbed the flight before take-off.
An assessment such as "2 meter visibility" would certainly make one think about going to one's alternate airport, as when one comes to the Visual Descent Point or Decision Height, 1) the visibility will need to be significantly better to be able to see the runway, or 2) it'll be a go-around or diversion at that time. One needs to prepare oneself dispassionately for the best and safest choice at that time, avoiding a terror-driven or get-home-now-we're-so-close response.
But 2 meter visibility at 2000' doesn't necessarily preclude breaking out below the clouds at 300' AGL four miles ahead. PIREPs (Pilot Reports) are oh, so welcome at such times, as well as a recent RVR (runway visual range) report.
HF