If that works ok then it would suggest the smoking gun is the itinerary of the ticket being incorrectly checked as a restriction.
And believed by Customer Relations to be a restriction, so much so that they're telling passengers it is.
This needs to be nipped in the bud, because in reality, except for those willing to get their elbows out, the restrictions are whatever they're believed to be and enforced as being on the ground.
It is possible to make a connection between the Charing Cross platforms and the Cannon Street platforms in 3 minutes flat, without running, but the MCT is 15 minutes. The idea that holders of off-peak tickets must adhere to the MCTs is wrong. The idea that they must begin their journey at the origin station on their ticket is wrong. They could begin the journey at London Bridge.
I can understand that the TOCs are going to want to use scan and tap data to enforce peak restrictions, yet they're trying to do so in a really crude way that demonstrates why they can't really be trusted to hold and process this data responsibly. If the ticket was tapped in at the origin station, or any station along a valid route (i.e. starting short) at a valid time, then the exit barrier shouldn't be rejecting it full stop.
I can just about understand if there was no tap in and the exit takes place after peak restrictions have ended, but before the first service arrives on which an off-peak journey from the origin could have been made, that you would want the barrier to reject the ticket for manual inspection. Arriving sooner than you could have got there from the origin doesn't prove you travelled at peak time, because you might have started short somewhere ungated or where the gates were open. But I can see why they'd want the passenger to be asked where they got on, and maybe investigate if it looks sus. But it is inappropriate to use the MCT to calculate which is the first service the passenger could have used.
You're only looking to rule out physically impossible connections, not any connection that a journey planner wouldn't offer as a copper-bottomed guarantee because it's a bit tight. Especially where the MCT is 15 minutes but the connection can be made in 3.
Edit: I do understand the TOCs are doubtless constrained by the logic they can use to program barriers without incurring enormous costs. What I'm saying, though, is you can't substitute the elegant logic you want for something more crude if the effect is to stop people doing something that's reasonable and commonplace and permitted. Instead, if the elegant solution isn't possible, choose something crude that is more permissive (like accepting off-peaks after peak restrictions have ended).