No need to go onto the Isle of Wight for a week. Your kids will be bored to death.
Go there for a day & drive round it instead (should be about £35 for a day return)
Portsmouth Historic Dockyard is best done over two days I’d guess (the tickets are valid for a year anyway), Jurassic coast is awesome (and ‘free’ if NT)
Maybe stop somewhere just west of Southampton as a base in the New Forest, that way you get the best of the road network and can get about