I live in South London and know the climbs you’re talking about very well. In fact when I started cycling they regularly defeated me - I remember walking up Chalkpit Lane, pushing my bike and feeling very unfit and ashamed. A few years later, they now hold no fear for me, my Strava positions are respectable, and I even enter the Bec CC and Catford CC hill climbs, though I’m still some way off worrying the podium.
And if I was to give my most honest advice on how you can improve going up them, I’m afraid it’s simply this: get fitter.
Unless you’re planning to drive to the climb, go up it, then drive back, you’re always going to be partly fatigued before you begin the segment. Especially so if they’re coming in the middle of a sportive like King Of The Downs. Just getting to the bottom having expended fewer matches is going to make as big a difference as (for example) slightly increasing the percentage of your FTP you can do for 3 minutes.
So don’t worry too much about particular workouts. Just follow a plan and keep the FTP going up - and if you can, the weight going down. Join a club if you haven’t already (advert: if you’re in South London, Dulwich Paragon are great!). Going on group rides to those climbs with people at or just above your ability is a great way to get better at them.
You’re doing the right thing already. Like @mcneese.chad says, General Build is a good mix of workouts - so keep at it, and the rewards will come.
(The only exception I’d make with workouts is that you might consider including a few low cadence intervals. At 234 watts for 75KG, a 20% gradient is inevitably going to involve a bit of grinding.)