Day 1 Lyme Regis -> Weymouth Corfe Castle / Durdle Door /Lulworth Cove/Lepe Beach-
Stay around Bournemouth
Day 2 New Forest ->Winchester->Jane Austen House->Portsmouth
Stay around Portsmouth
Day 3 South Downs->Trundle -> Seven Sisters ->Hastings /Battle /Rye/East Dean
Stay around Hastings
Day 4 Dover / Canterbury
Stay around Canterbury
WoW
Do you intend to visit any of the sites in detail, or is it an instagram checkbox drive?
Day 1 - arriving from where? Not from an international airport after an overnight flight? Which of the 4 places do you expect to do in detail? Before going to Lyme Regis you should be aware of the tides. Are you going to The Cobb for the Jane Austen connection. My personal experience is that getting parking anywhere near is non-trivial.
Day 2 - 2 major cities and Chawton. Do you intend to visit both houses in Chawton? Check opening hours and days, they are not always open all day, and both are not always open at the same times. My experience is a minimum of an hour at the smaller property (we take 2 or 3), and at least 2 in the bigger - it is a big property and the carpark is a bit of a walk. Both are excellent and full of experiences.
What do you want to see and visit in Winchester? There are lots of things to see in the city; the water mill and Round Table get good amounts of attention to say nothing of the Cathedral.
Similar question about Portsmouth. Will you visit the ancient ships at the Navy Yard? Go up the Spinnaker? Hydrofoil (or traditional ferry) over to the Isle of Wight?
Day 3 - wow, that's a lot of driving. The south coast is not fast driving.
Day 4 - What do you want to do in Dover and Canterbury? Traffic in Dover can be AWFUL. Remember that there is a huge amount of cross channel traffic going to and from the ferries, and some of the signage is not absolutely clear. Parking can be diabolical in the town, easier inside the castle. The roads are narrow, there are double yellow lines everywhere. Much of the cross channel traffic is heavy trucks, some British, much European. European trucks will not be able to see as well because the driving position is on the wrong side. BEWARE.
Are you back to the airport on Day 5? What time is your flight? Which airport? Canterbury is not at all convenient to any London airport, and if it is Luton or Stansted you will have the very slow moving tunnels of the Dartford Crossing.
Are you familiar with Operation Brock?