When you say that you will 'emphatically resist any attempt to recreate a London where the rich and poor cannot live together,' you show yourself to either be a prime BS artist or an extremely out of touch toff. Someone receiving housing benefit capped at £400 a week is still living in a home that is unimaginably beyond the reach of most salaried Greater Londoners.
If you live in a house worth £1600 a month in rent, it makes no difference whether that £1600 is first paid to you as part of your salary before going to your landlord, or paid directly to the landlord by an employer, a wealthy aunt, a trust fund or the government. Either way, the value of the benefit you're receiving by living in the £1600 house means you are not, by any stretch of the imagination, 'poor.' And of course, the people living in those £1600 houses are unaffected by the planned cuts - the only people struck by this 'hardship' will be those living in properties costing more.
By all means, mitigate or make exemptions for the little old ladies for whom the disruption would be especially cruel, but for everyone else, the disruption, while unpleasant, would be no worse than for anyone else who's had to pack up and move away to cheaper areas over the past few decades.
Imagine how many genuinely poor people could be helped with the small fortunes the government hands over to inner-city landlords every month.