Of course if delay repay was centrally administered then the passenger wouldn't end up in the ridiculous situation of being delayed but not being able to easily claim.
Presumably it would be a lot cheaper to administer as it would cut out those disputes where the receiving TOC doesn't dispute that the passenger was delayed and is entitled to claim, but insists they must claim from a different company. The system of forwarding claims is supposed to protect the passenger from the administrative burden of this, but it's still an administrative burden for the TOCs (and we do sometimes hear examples of claims being forwarded wrongly).
And it would mean we wouldn't have to worry about these edge cases where it's very difficult for the passenger to know which TOC caused the delay and, therefore, who to claim from.
In fairness this one does seem like an unusual situation. It sounds like rather an unsatisfactory setup - presumably it must be a fairly regular thing for passengers to arrive there a couple of minutes before their train is due to leave only to be prevented from crossing until their intended train has left the station? Or, if train movements are such that the barriers stay down for a long time, people could arrive rather earlier than that and still miss their train? And presumably the industry simply blames the passenger for not being there earlier?
In this case the OP arrived at the station 7 minutes before his connecting train was due to leave when the MCT is 5 minutes, so any suggestion that he ought to have arrived earlier would be nonsense, of course. It'll be interesting to see what happens.
I think with both trains being on time you are going to be hard pressed to get either train company to cough up for Delay Repay. If anyone is liable in such unusual circumstances it should be the operator of the 'heavily delayed' train.
If it was a passenger train. I don't think the "you claim from the operator that caused the delay" rule is a very helpful one in situations like this. It really shouldn't be the passenger's problem who's fault it was and, frankly, the industry spends way too much time arguing between its different fragments about this sort of thing.
But assuming it is was a passenger train then I agree the OP could try claiming delay repay from the operator of the train that was actually late, but as that wasn't the train the OP travelled on or intended to travel on, it's going to need manual intervention via somebody in customer services as Hadders mentioned.
Sounds like the minimum connection time needs to be increased if the first train arrived on time and the connection was missed.
What would the impact of that be, though? Increasing the MCT would make connections such as the OP's that are currently possible impossible, which given some routes through the station have a three-hourly service, would go a long way towards making rail unviable as an option even where it's just about viable today.
Yes, in this case a late running train made what would otherwise have been a comfortable connection impossible, but I don't think making that connection officially impossible all the time would be an improvement. A better solution might be to see if the stop markers could be moved (which may require moving equipment and maybe even signals and track circuits, admittedly) so that trains don't block the crossing when dwelling at the platform?