At Delta the 737-800 can actually be handflown to Cat-III with one or both engines running.
Why would you want to (engine out)?
First of all the HUD works great. As part of checking out in the airplane when we first got them we went out and shot touch and goes with the flying pilot's outside vision obstructed. Many takeoffs and landings on a relatively windy day and all were right in the touchdown zone and right on center line, every time. This was a great confidence building exercise.
The HUD allows for much lower minimums for takeoff. So the problem arises, what do you do if you lose an engine during a verly low vis takeoff? The answer WAS you HAD to divert to an airport with CAT 1 minimums. Now with approval to shoot CAT-III with an engine out we at least have the option of landing at airport of departure vs. a long divert on one engine.
This also allows greater dispatch flexibility since you don't absolutely require a take-off alternate during low-vis ops. (As always the captain may require more wiggle room than the absolute letter of the law).
Also, the 737 A/P is two axis. So the pilot must make all rudder inputs. And in any handflown CAT-III approach the auto-throttles can't be engaged. And the A/P flare program is not setup for engine out/ flaps 15 etc. etc. Long story short, it only really works to hand fly an engine out approach, whether it's VFR or CAT-III.
Personally, with 4 years experience with the system, I wouldn't hesitate to shoot a hand-flown CAT-III approach engine out or both running.
BTW the HUD equipped 800 can be dispatched to CAT-III minimums with the A/P inop. The HUD really gives us a lot of operational flexibility.
The crazy thing is in four years on the airplane I haven't even shot as low as a CAT-II approach. The one time I thought I might get to shoot a CAT-III the weather was below CAT-III mins and had to divert. On the MD-88 I shot several CAT-II and CAT-III approaches.