I'm very interested to hear what you think.
I suspect ordinary Londoners may have been more stoic than much of the ruling class
Remember that I'm coming at this from the East End, which was very definitely not the ruling class part of society. The East End boy also has a line in cynical grumbling that is unparalleled - praise is not common, and scepticism is a way of life. These attitudes certainly continued well beyond WW2, and the West End mob (ie, the rich and wealthy and powerful) didn't regard East Enders as being quite human, and certainly not people to take into consideration when making decisions.
To give an example from later years. In the late 1950s, a young East End girl (aged around 16) was raped and murdered and dumped. On being informed of this, the police regarded this as a big joke, and had no intention of doing anything about it. "It was just a Dagenham girl. It's not important."
So bombs (of whatever variety) dropping on the East End might cause problems for the people who live there, but for them in the West End, it's just a bit of slum clearance. Only important in terms of the impact it has on things like the docks.
Point No 1. "Whitehall/Westminster will bow to German pressure if the East End is being hit repeatedly" is complete and utter hogwash. The problems of the East End would only be important in so far as they affected the West End.
Point No 2. The East End took the brunt of the Blitz, it took the brunt of OTL V1 bombs. It adapted to life (and death) and the circumstances it was in. Not without cynical black humour, it has to be said. A fair amount of the ordnance that fell and failed to explode was recovered locally and repurposed. There wasn't a bank safe that was secure in the immediate post-war environment. Stoic isn't quite the right word to use to describe an East End boy - stoic implies uncomplaining, and until you've heard an East End boy, you haven't heard complaining. But adaptable, yeah, that's a good word. The situation is what it is, and you make the best of what you've got to hand. If that involves stepping on the wrong side of strict legality, well, them as upholds the law weren't on our side, so there weren't no reason for us to pay heed to laws that weren't to protect the likes of us. This, remember, is the background from which the Krays came (truly nasty pieces of work lionised by the Powers That Be).
Point No 3. Tunnels. You wouldn't believe how much stuff is underground in the East End. Not just the Underground (which in the true East End is largely on the surface anyway, though it's still called the Underground), but any number of tunnels. There's a whole ecology there, and if houses on the surface are being destroyed, life will go on anyway. It'll change and adapt. That's what Londoners bleeding well do. Adapt.
Point No 4. Payback. Traditionally, we have long memories and retain grudges for long periods and "Payback" is part of the culture. We hold grudges, and we'll look for payback, big time. If London gets bombed like that, well, things like what happened to Dresden are on the cards. Incidentally, the East End response to what happened at Dresden can be summarised in one word. "Good." The local intellectuals might follow it up with: "Shame it was just Dresden." Bomb us, expect to get it back in spades. And you can expect that attitude to feed upwards.
Point No 5. Militarily, bombing London is a bit pointless. It doesn't have airfields in any great number, the ports and industry and so on is all replicated elsewhere. Bombing London is a purely political attack and will achieve zilch militarily. One is expended whatever resources are spent in lobbing exploding bricks into the area in order to knock down houses that the Powers That Be simply don't care about. It's a waste of effort.
Point No 6. Read the memoirs and diaries and stuff of the people that went through it OTL. There's enough of the stuff. Mass Observations is a good place to start, where ordinary people recorded their thoughts at the time. There are tons and tons - literally - of books on the subject. Writing about such things as this without going through such records is like trying to write about life in the WWI trenches without going through the Trench Newspapers. I have been greatly amused by some of the pontification about what people of the time thought without any apparent research into what the people of the time said at the time. That's not how history is done. If you want analysis of comparative weapon systems and balance of economic resources and forgone costs, that's fine. That's a valid line to take. However, if one talks about the views of the people at the time, it is incumbent upon the historian to find out what the views of the people at the time were, not what they imagine they were.