It seems that you are doing a lot of price avoiding. In order to do that you are backtracking and going way out of your way to save some money. But all that backtracking and using Birmingham Airport to avoid London air prices can cost a fortune in extra rental days or train tickets, to say nothing of the time value of money. Or put another way, you may find you are wasting time - your most valuable commodity, especially on such a short visit.
If you need an extra hotel night in Birmingham to make a morning flight there, what does that do to your budget? How much are you budgeting for the train ride? Slow and cheap on London Midland or fast and expensive on Virgin?
If you land at Southampton why not do Bath which is quite close before you get to London? If you have to backtrack how much of the train fare (you don't want to drive in London or Bath, trust me - especially Bath on the opposite side of the road with all their steep narrow hills, parallel parking and manual transmission) could have you spent on a nicer London hotel or one which fits your schedule?
May 1 is May Day, May 2 is the Early May Bank Holiday. That weekend is likely to be crazy busy with traffic both on the roads and on the trains. And trains may be disrupted by holiday engineering works - they usually are.
Remember that many speed limits are not posted and you need to know them based on the conditions. Many times the information you need is painted on the road - sometimes clearly, sometimes not - and you may or not have supporting signs as you are used to.
