IANAL but does this argument not smell like Bundling?
https://en.wikipedia.org/wiki/Bundling_(antitrust_law)
It's not quite the same because Bundling seems to be about setting a price that is too low, so this isn't a legal precedent but it seems like a morally similar argument?
You are saying that you paid Apple for a device, and that entitles you to access to their services (e.g. Apple bundled their services with their device for free).
If this was Apple's main contention with Beeper, then why not allow people to pay a one-off fee to access their services (whatever fraction of the retail price of a device goes towards services)?
I suspect they can't do this because then you would be fully in Bundling territory: that the one-off price you pay to access Apple's services becomes cheaper when bundled with a device.
The whole idea that you paid a one-off fee and that covers your indefinite access to Apple's servers at a continuous cost to them also seems like a flawed model if that's really what the economics of iMessage are. Of course the contention is that the continuous fee to you is the fact that you are locked in and are therefore helping Apple attract other people to their ecosystem.
There is also another argument going round that somehow the issue is that Beeper is charging for their service. I don't buy that, because I pay for Beeper and I don't use their iMessage integration (because I live in the UK so this whole blue bubble green, bubble thing is just not a thing here and we can just enjoy watching it unfold with idle curiosity) and I pay the same as someone who would. I am paying them on a continuous basis because they are running servers that I want access to that bridge to other services I use and want aggregated in one place.