I would say that you (zenten and Katarn) are both right.
Fact: BCVs are subject to the Living Dead power, which prevents natural healing of physical consequences. Supernatural healing methods work normally, and this power implies that there are other, semi-mundane ways of healing (with taxidermy mentioned specifically), but no mechanics are given.
Fact: BCVs also have the Blood Drinker power, which grants them a supernatural means of healing minor physical consequences.
So BCVs can't heal physical consequences by the normal method (ie, wait out the recovery process in scenes or sessions). They have a specific supernatural power to heal minor physicals; this bypasses the Living Dead restriction. In addition, OW86 'extrapolates' that BCVs have a "rejuvenation by blood" power.
As one last consideration, the Blood Drinker power doesn't actually specify that minor consequences are healed; rather this is a specifically noted benefit of the actual power, which is to gain (supernaturally) a scene's worth of recovery, once per scene, by killing a subject through drinking their blood. So one could extrapolate this to say that so long as a BCV drains a victim every scene, they gain the supernatural equivalent of normal recovery.
Here's my suggestion on handling them. Consider these house rules, and take 'em or leave 'em as you see fit.
BCVs don't heal normally, but they can be repaired through a combination of physical rebuilding of their bodies and regeneration powered by the deaths of victims. First, create a new stunt called 'Corpse Repair' (or just 'Taxidermist, if you prefer) which allows a character to use their Craftsmanship skill to repair structural damage to corpses (but does nothing to heal the living). Characters with either this stunt or the 'Doctor (Medical)' stunt can repair structural damage to corpses, including BCVs. The dificulty of this task is based on the level of the consequence being repaired; for example great for moderate or fantastic for severe. (Note that these difficulties are noted on YS220 for medical rolls to trigger normal recovery.) If successful, this allows the taxidermist to 're-label' the consequence to something less severe. For example, if a BCV was given the Severe consequence "Arm lopped off", it could be changed to "Damaged arm". This is still considered a severe consequence, but changed what compels can occur (ie, the vampire has gone from zero use of the arm to not-entirely-reliable use of the arm).
To actually recover from the consequence, the BCV needs to feed on death fairly regularly. How often requires some judgement, as 'scenes' can be variable lengths of time. A minor consequence is easy, since it requires one scene, therefore a single death, and this occurs nearly instantly. A moderate consequence requires a session (or one or more days, per YS315) worth of recovery. I'd go with, say, six 'rest' scenes ~= two days ~= one session sounds about right. A severe normally requires several weeks to recover from, so this would scale up to dozens of deaths. Note that the vampire can't simply "power-heal" by draining large numbers of mortals in one sitting; healing still takes time. This time is greatly reduced if the BCV happens to have a Recovery power, though I'd tend to favor ruling that the same number or deaths are required, but that the BCV can metabolize them more quickly.
As I said, these are ideas not RAW. Any thoughts?