No.
Just no.
If that is your plot for a game, I guess it is possible...but...
I know they could make a lot of vampires. I just think there is a learning curve on powers. The new vampires would be weaker and not know how to use their new powers to their full potential.
I think "It's My Birthday Too" mentioned something about the new Black Court vampires being very dangerous but not used to being full powered vampires. That was why Harry was scared but not friggin' terrified.
(White Wolf Vampires seem to work that way and I always liked that.)
I'm pretty certain, that if we're following canon, it isn't possible. If it were that simple the scourges would have been hunted to complete extinction. Barring that, the entire world would be a human farm for BCV's. Some ambitious vamp would have tried it and found it it doesn't work.
However, with a book like the Word of kemmler..... that Harry gave Mavra, could start a BCV Renaissance : the Black Court War. Which I think Jim Butcher had/has plans for in upcomming novels.
Besides nothing is unbeatable. Kemmler is beatable. Get a few KoC's together, have them all drop a Righteousness Nuke. Problem mostly solved. (I assume most BCV have lower disciplines, most BCV newbies should have a lower discipline for sure.) Sure, that leaves plenty of the older tougher ones or unique younger ones, but it is far from zombie epidemic.
The fact that the world hasn't been completely overrun by vampires indicates that vampires don't have disease level reproduction otherwise they would win one vamp could easily create dozen more who could create dozens more etc an never wave of super powered death, KoC can't fight dozens of black court vampires they are only humans and wizards would be over run, cities could be overrun in days where talking end of days territory. So given they don't have disease level growth how fast to people think they can reproduce? (White Wolf Vamps don't have that level of reproduction either siring kindred is a big effort on the part of vampires).
Punctuation.
Sorry. In all seriousness. I think you already decided the answer to your question. They can infact do it. No one can stop them. They win. It is the end of days. This is goingto sound rude and to some extent it is. I have to ask though. If you seem to have the answer already decided in your mind...why ask our opinions? I can tell you I agree and let you hear what you seem to want...but I'd be lying.
Also, Knights can totally fight dozens of vamps. That is what they are for, saving the world from evil. They have Archangels backing them up. Wizards proved they can kill hordes of vampires in a novel called Changes.
I even just explained how to do it as a Knight. Been there done that. (BCVs and RCVs) Random people made into vampires do not turn into martial arts masters like in Buffyverse. Their skills will not match what combat veterans can do. A measly +1, +2, or, +3 combat skill will not match up to +5 or +6. I've seen teamwork in action before also in this game. It doesn't work quite as well as it sounds.
I answered your question fine.
The setting (assuming you ascribe to that and are asking about our opinions with that setting and canon in mind...) pretty much couldn't exist if BCV could make armies of powerful vampires.
Keeping with canon and setting established by Jim Butcher and the setting used by Evil Hat to create the DFRPG - I give you this explanation:
If vampires could do it; they would have by now. The fact that they haven't means they can't. I give vampires enough credit to think of that on their own.
I could see them making as many people as they could bite in an evening (assuming that is how they make more). I just don't think it matters or that they are of any sufficient quality.
Again I draw attention to "It's My Birthday Too". Several vamps were made in that short time frame. It can't take long, but they sure didn't seem very powerful.
In your own setting they can be as powerful as you want. They can reproduce as fast as you want. They can be Bruce Lee as soon as they get made.
If you want me to come up with a sane arbitrary number? I'll just go with Sanctaphrax's idea. Or let them make thousands and rule the world. Your game, your setting, your rules.
As an aside:
V:tM has rules for mass Embracing via the Sabbat Ritae. The vampires come out by the droves. They suck compared to most vampires. Seriously, if you make an army of vampires you can't be picky and usually get busses of cheerleaders heading to football games. (Or maybe that was just my favorite for shovelheads)...
I'm not sure if this was word of Jim or just something that was tossed around here but, I think there are three types of BCV. The first is the just above zombie level of functioning (the ones Mavra spoke through and they can't spawn others)
, the higher level BCV can spawn the lower level BCV but can't make a BCV that can function without guidance , then we have a master BCV thst can not only make other BCV but can make other thinking BCV and other masters. A BCV of the second level could spawn a large number but they are of very limited use. Masters are too smart to make a huge horde and get noticed. For the game I would say there should be enough first level vamps at any one time to pose a threat to the group and unless they get the higher ups they will just replace them. I would not say that the scourge would not grow exponentially because either it can't be of much use(level two BCV) or it won't (master is to smart).