Engineer,
Indeed, for a BFL takeoff, TODR=ASDR.
Perhaps you are confusing the (runway) available with the (aeroplane) required distances ?
BFL relates to the aeroplane, not the runway.
A runway may well have TODA=ASDA or, as is more often the case, TODA>ASDA. This has not a great deal to do with a balanced field length takeoff.
However, if the aeroplane AFM calculations, for the conditions pertaining to the takeoff, are scheduled with a selected V1 (or V1/VR - whichever you prefer) such that TODR=ASDR, then the takeoff is said to be scheduled on the basis of a balanced field length. You can, as a result, have a BFL takeoff scheduled on a runway which is several miles long and has a great mismatch between TODA and ASDA... the aeroplane is not particularly interested in all the spare runway which might be available.
If clearway be declared such that TODA>ASDA, then the BFL scheduled takeoff cannot take advantage of the clearway's "extra" distance. To do so would result in TODR>ASDR.
The V1 scheduled for the takeoff is determined from the interaction of a number of competing AFM cases which have to be assessed on a takeoff by takeoff basis. Generally V1 will only be driven to the maximum if you have plenty of runway available to play with. For the BFL case, the numbers will depend on the specific aeroplane and the specific takeoff conditions on the day. I don't think that it is valid to try and generalise... that is to say, for a BFL calculation on several aircraft, the V1/VR takeoff speeds scheduled will, almost certainly, be different.
However, if I have not satisfied the thrust of your question, please do restate it again as you see fit. I am quite sure that many people are not comfortable with what we performance chappies do for a crust (and how we do it) and it is probably worth exploring the subject for the greater benefit.
In the end analysis, though, I wouldn't worry too much about balanced field length calculations. Quite apart from anything else, the great majority of takeoffs are very definitely not scheduled on that basis ... otherwise the accountants would be screaming for the lost kilos.
The only significance of a BFL takeoff is that the AFM usually provides some simplified charts for the BFL case which means that the runway limit can be determined just a tad faster than going through the unbalanced calculations which are more general (and commercially useful).
There is no general advantage in scheduling a BFL takeoff as, on most occasions, the RTOW improves if the takeoff is unbalanced.
Have I helped ? ... or hindered ? .... or have I merely confused myself ?