To me that facts that led to him dropping down through the sucker-hole are interesting in that they led to that decision. Part of the investigation and something to learn from for sure, but separate from what happened down below cloud.
Once through the cloud layer to go into Battersea there are probably three competing pressures; at this point I discount the pressures (self-imposed?) from above the cloud to do the best job he could for the client. Once heading down through the cloud-hole he was concentrating on putting the machine down at Battersea.
The first pressure he faced when he popped-out VFR below is noise abatement. Weather so bad, why not hover over the river and get cleared in? Not ideal from a power point of view, but solo in that machine? No problem! There is a circuit pattern at Battersea designed in part for noise abatement and hovering over the river whilst waiting for clearance would be legal (rules of flight) but generate complaints for Battersea. Safe, but very unpopular. Did it even come into his mind? Perhaps a low-hour pilot would think this way, but then a low-hour would not have been there.
So he flew the circuit
After that, as said already, he has pressures to remain +500ft from anything and to stay VFR. My guess is he was trying to do both which led to a climb and ..... he broke the 500ft rule.