Touring World War 2 sites/museums in southern England advice?

You might enjoy the Tank Museum in Bovington. Have a google for exhibit specifics and running days.

Hire car is the easiest way to get there.

I would imagine that a car would give you much better access to sites etc than public transport.
In Dorset if you have a car you could visit the tank museum in Bovington and Tyneham a now deserted village that was requisitioned by the MoD during WW2. You have to drive over the military ranges to get to Tyneham which is an adventure in itself, look out for signs warning of unexploded shells!, and the road is only open not surprising when the military aren’t driving their tanks over it and practising their firing!

While not WW-II, the naval yard in Portsmouth is worth a visit. It has an interesting history and the HMS Victory. There are the WW-II tunnels at Dover, and if you are interested in cryptography, Bletchley Park at Milton Keynes is nice. Also north of London is the Imperial War Museum at Duxford. It is one of only a few places where there are flying Spitfires. Another is Biggin Hill, south of London. Further north from London, near Uppingham, is the old Spanhoe troop carrier command airfield (privately owned, but you can walk around). While the tower is gone, the taxiway and runways remain, as do some hangers. Many paratroopers departed from it headed to France during the D-Day invasion. With only a week, my choice would be a rental car so you don’t have to worry about bus or train schedules.

I heartily recommend the Bovington Tank Museum, unlike anything else, extensive, tanks of the world wars, and post 1945, a huge place. Taxis are available at Wool's train station. I take the train from London Waterloo to Wool, the closest stop to the Museum. From there to the Museum cost 7 GBP In your case rental car may be best in reaching the Museum. Prepare to spend the day there.

There is also the D-Day Museum in Portsmouth. along with the large WW1 memorial.

Dover castle itself is well worth a visit, because although it dates back to Roman times, there is an extensive network of tunnels that served as an operations centre and hospital in both WWI and WWII.

"While not WW-II, the naval yard in Portsmouth is worth a visit."

The Naval yard itself played a huge role in WWII as one of the major naval bases in the UK. There is a large amount about WWII in the Royal Navy Museum which is within the dockyard complex. While definitely not WWII era, the Mary Rose exhibit is one of my favourite museums anywhere.

Just over the hill from Portsmouth is the village of Southwick, where the main planning centre for D-Day is located.

If you are arriving in Dover, a great nearby museum is the Kent Battle of Britain Museum near Folkestone run by enthusiastic volunteers, who are experts on the topic.

A car is the best way to get around.