I'm not an airspace construction expert, but you need to be aware that around LL/KK/GW/SS there are many conflicting routes which have to have some form of procedural separation from each other in the case of R/T failure.
To give you an example, there are many departure routes over BPK. The highest is the LL BPK SID which places an aircraft at 6000' over BPK. Then there are SS/GW/WU departures which occupy the levels from 5000' down to 3000', but I can't remember in which order. WU departures are sometimes struggling to make 3000 ft by the edge of the LL CTR anyway. Don't forget LC departures as well, which have to remain inside CAS (base 2500') but remain separated from the other traffic.
Then there are the arrivals which have to descend into the various airports. Minimum stack level is the lowest FL which will give 1000 ft separation against 6000' QNH (usually FL70 or 80). To raise the base of CAS would push all the SID levels up and the stack levels as well, making it difficult to lose the height in time to land.
So in short, it has little to do with aircraft performance and more to do with route conflictions. Don't forget the Worthing CTA base was raised a few years back because a/c performance had improved such that the lower levels were no longer needed.
The airspace around LL CTR is indeed funnelling a/c into a very small volume of airspace, and the lack of a reliable LARS service between the LL and GW CTRs is in my mind an accident waiting to happen, but that has been discussed in depth before on this forum. All I can say is: KEEP A GOOD LOOKOUT!! By the way, Northolt Approach are most helpful in providing a service around the NW corner of the LL CTR.