I wasn't aware that the fail-operative 737s were CatIIIa on one donk, though it makes sense. That being the case, Denti is right that you wouldn't need a take-off alternate provided the departure airport was above CatIIIa minima.
The 777 is CatIIIb on one engine, and as such we don't require a take-off alternate unless conditions are below the best approach available, up to and including 75m viz (in which case we couldn't take off anyway!!)
Interestingly, we also have the stipulation that the take-off alternate aerodrome can be 2 hours away (except in the US), provided we are in an "ETOPS Aircraft", which is interpereted as meaning an aircraft certified for ETOPS, even if we are not planned and signed out for an ETOPs sector.