If you don't need the vehicle on a daily basis, consider parking/storing it further out in the 'burbs, near an overground or tube station. It'll be cheaper than trying to find parking in or near SE1. I have done this myself with success.
I am in the same boat, I live in Coin St and need a lock up garage or a off road space somewhere to park my car. I currently rent a space in the ACP carpark in the National Theatre, but it is costing me a fortune. I would need somewhere buy the end of september.
I use the car only at weekends and so somewhere in the vicinity or walking distance of Coin St would be great. Any offers?