The old school v.s. Extended range is slippery terrain, indeed. It broke into fights before, so we better not go there

People have their reasons for wanting the particular instrument they have, and I think any musician should be able to respect that.
As for the question at hand, well, it it is the number of strings that is a problem, then yes, it is possible to tune to BEAD and have the same number of strings with access to the lower register of a five. However, if it is the SIZE that is the problem, it becomes a less viable option...
As people said, to get a good tone out of the B string, you would need to use the bigger gauges (I would say .115 as a minimum, but anywhere from .120 to .130 would be preferable - mind you, this is my own unprofessional and very humble opinion) . You would aslo have to take into account that depending on the bass, a tune up would probably be required to make sure the string fits (especially in your nut) and has the right tension (a strange fact, but higher gauges B strings usually exert less pressure on the neck than the E string). You might want to intonate too, if you want to use this tuning a lot.
It might sound like a bit of work, but it can be done - professionally, if my memory is correct, Justin Chancellor from Tool used the BEAD tuning frequently. It's all a matter of what feel you want, in the end... Although I fear to get a good tone from a low B, higher gauges a pretty much unavoidable, IMO :/
Hope this helps...