If you think a little bit laterally, you don't have to go to CJ or Waterloo.
You could cycle to Surbiton (about 5 miles from Twickenham) and you will avoid the London-bound restrictions. You can also get more of a lie-in, as you can get the 8:44 from Surbiton, change at Woking and Basingstoke and get to Bournemouth for 11am. Or, if you get the 8:30 from Surbiton, you only need to change at Woking and you are in Bournemouth for 10:20.
Return ticket from Surbiton is £35.80 which, by the time you've factored in your other car costs, will probably be pretty similar to the cost of driving.