If you take a look at their How it Works post [1] this is not an entirety client side implementation, so there would presumably be a small number of IPs that would need to be blocked.
So your point is Apple can presumably distinguish between an actual iOS connection and Beeper's connection by looking at "how many connections per IP"? Still seems prone to false positives to me, unless there is something else I missed.
(Upon re-reading the post, I realized that the phone number registration is actually done by Apple. Wonder if this might provide another basis to block Beeper, i.e. all this SMS infrastructure is not cheap to maintain and Beeper's integration is arguably using it in an "unauthorized" way.)
They don't need to break it completely. If Beeper is unreliable, nobody is going to pay for it.