Hardcore (rubble, not the other sort!) is difficult to get rid of legally, and the Lewisham depot will send you away with it and ask you to go to a contractor nearby who will charge you by weight, weighing your car on a weighbridge.
Try the new Southwark Council facility just off the Old Kent Road: lovely undercover space, easy unloading as you drive up to a high level and drop the stuff over a wall into a bay. Lots of my rubble has ended up there and I have not yet been asked for ID although I do have ID as I pay Council taxes in Southwark. Just about anything accepted and helpful staff.

Don't forget that small electrical/electronic equipment can go in the metal bin in Sainsburys Car Park, FH, or at Laurence House Catford.
WEEE stands for Waste Electrical and Electronic Equipment (Directive) which is all about proper disposal and recycling of such stuff.

But don't WEEE in the bin!