If you can keep the bikes in the room you can probably find somewhere to park the van in Southwark but it won’t be cheap and you’ll need to be out of London very early to avoid traffic and allow for the joys of M25/M23.
If you do change hotel the Novotel in Greenwich has underground parking and is pretty easy to get to from Wembley arena. Also can be on the A2 easily for a sharp exit in the morning.
If you park at Gatwick the trains from London Bridge start early so you should be able to get there, get the bikes and check in.