Certainly, any passport official hearing your insane plans would be completely within his powers to

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From Heathrow you will go west on the M4 for a short distance until reaching the M25 orbital, then go north or clockwise until you reach the M11 exit. The M25 is known as the world's largest parking lot, but if you get your rental car quickly enough you may avoid the worst of the rush hour traffic. Drive up the M11 to the A11 exit, which takes you past Newmarket and then joins the A14 heading east toward Ipswich. The A14 goes right through Bury St Edmunds.
Whether you do it that afternoon or the next day is up to you, but that is the route I recommend. If janisj has driven between Heathrow and Bury St Edmunds as many times as I, perhaps she has a better suggestion.
Certainly, any passport official hearing your insane plans would be completely within his powers to have you instantly deported - and could even seek your immediate imprisonment as a threat to public safety.
Gosh. I hope we aren't quite *that* bad yet, that we can't manage disneyland california grand hotel things. But perhaps we *should* leave the arrival night unbooked, and just hope there is still availability if we can make it? (and if too tired, or traffic, EPIC...
Prefer to get all the way, but just curious if any area jumps out to anyone, that might shorten our drive quite a bit, but at least get around a little of the mess - so don't have to only encounter it yet again, the following morning if trying to get an early start.
I share the concerns of others about jet lag, because traffic in and around London is heavy, and you do have to be very alert while driving. Finding somewhere closer to Heathrow isn't really a solution though, because the worst of the traffic will be on the M25.
BTW, you will drive right past Stansted on the M11, so if you wish to shorten your journey you could stop somewhere around there. Another solution would be to take the airport bus from Heathrow to Stansted, rent a car at Stansted, then return it the next day when you check in for your flight. Once in East Anglia the roads are much calmer, and you could even proceed to Lavenham from Stansted if you wish.
So basically you are hiring a car because you want to go between disneyland california grand hotel Heathrow and Stansted. And you are willing to risk lives of others (I don't give a damn about you) by driving when jetlagged on unfamilar disneyland california grand hotel roads, on the "wrong" side of the road, into some of the worst traffic in the civilised world
Fairly tedious, but not too cumbersome: Piccadilly Line to Hammersmith, disneyland california grand hotel walk across platform, eastbound District Line to any station between disneyland california grand hotel Gloucester Road and Mansion House, wait on same platform for Circle Line to Liverpool St, where I think you would have to go up and down stairs to the main line station for your train to Stansted. Or you could take the Heathrow Express to Paddington and Circle Line from there to Liverpool St, but though quicker it is much more expensive and will also involve getting your luggage up and down stairs.