I don't remember the last time I saw Stansted or Essex listed on the CFMU regulation list when it wasn't for fog or wip or an incident.
I don't think that it is a reguar place where flow is applied to the airspace.
Even Heathrow is rarely regulated for the whole day.
You don't need regulations to demonstrate you are at capacity. Flow managers can use a variety of techniques which allow volumes of airspace to operate at or above capacity without anyone on a flight deck ever being any the wiser. These techniques can be tactical or strategic and generally would not involve light aircraft in any case.
On that basis the CAT driver with his ATPL does not have a right to enter, nor does the AOC holder who is operating the aircraft as all of these are privileges that are at the mercy of regulators.
The CAT will have filed a full flight plan, and provided the pilot remains in Controlled Airspace for the whole flight and thus doesn't require a join then the clearance granted at the departure airport is good all the way to destination. This is regardless of the fact that different ATC units are responsible for different parts of the route, the philosophy being that the profile of the aircraft will be co-ordinated ahead of it's passage. Clearance limit points might be issued along the way, or holding, but in the event of radio failure the pilot (and ATC) know what to expect. As they're scheduled customers of the destination airfield, they are unlikely to be refused entry in any case.
VFR GA could quite legitimately file a full VFR flight plan for a route totally within Controlled Airspace (e.g from Edinburgh to Glasgow) and they'd be treated the same as the CAT flight with a clearance and co-ordination taking place automatically.
VFR flights could also file VFR flight plans with transit units along the way if they choose to. Whether this will help is debateable since many pilots don't know how to address them correctly anyway, and many ATC units won't do anything with them even if they are received.
Stansted has had a reputation for many years of being very hard to get a transit clearance from. The problem is that much of it is anecdotal and perhaps seen as people just bumping their gums. What the CAA are interested in are factual reports which detail the refusal (date, time, details, etc) which they can then discuss with the ATC provider. As I said earlier, sometimes the provider can provide a valid excuse which the CAA accept. Sometimes they can't. What the CAA can't act on is a general opinion, either on an internet website or other forums, unless there is reported evidence which they can investigate.
The ball is firmly in the court of GA pilots who have been refused a transit.