Balanced field length calculations by definition are the greater of three factors. As Lnafziger noted in his answer to the linked questions above these three factors are:
- 115% All engine takeoff distance
- Accelerate-stop distance
- Accelerate-go distance
The manufacturer will choose a $V_{1}$ speed that will all the accelerate-stop and accelerate-go distances to be as close as possible. This reduced the takeoff distance to the lowest possible since the takeoff distance is the greater of those three factors.
This means that our takeoff distance must be less than certain declared distances. See the linked questions for a discussion on declared distances.
- The TORA value may or may not equal the runway length but it will never be greater than that length.
- The TODA is the TORA plus a clearway.
- The ASDA is the TORA plus a stopway (it can be lower than the TORA value).
Using common sense, our accelerate-go distance cannot be greater than the TODA value and our accelerate-stop distance cannot be greater than the ASDA value.
If we take this one step further, since we have one number for takeoff and we don't know if that number is limited by the accelerate-stop or accelerate-go performance we cannot use the TODA value for the accelerate-go distance. We must use the TORA value.
Hence, for multi-engine airplanes with balanced field length calculations we are limited to the lower of the TORA or ASDA values for our takeoff distance.
As Aeroalias and Lnafziger pointed out, for unbalanced field length calculations we can indeed use the TODA value for the accelerate-go distance and the ASDA value for the accelerate-stop distance.
References:
The NBAA has some great videos on the subject. Here is the video on YouTube.
AOPA also has information on the subject