This was actually a question the Royal Navy's designers often faced, especially as later iterations on the Lion-class design grew massively in weight, but retained similar armament; there were several 1945 designs with 60,000 ton displacements and nine 16in guns. Answering it involves looking at the detailed differences between the designs. To keep things simple, I'm using the 1939 design for Lion, rather than any of the later designs. This had a standard displacement of 40,550 tons, while South Dakota's was 37,375.

One main difference between Lion and South Dakota was in their physical size. Lion was a larger ship overall. It was 740 ft long, 105 ft wide and 58.5-54.8 ft deep, compared to 666ft long, 108ft wide and 52 ft deep for the South Dakota. Lion's longer hull made her more hydrodynamic, meaning that she was faster with less power. Lion's engines made 120,000 SHP, compared to ~130,000 for South Dakota, but she had a design speed 2.25 knots faster. This increased length, though, made her hull heavier. Comparing hull weights between British and American ships is difficult, as American designers counted the deck armour and torpedo defence system as part of the hull weight, while the British counted this as part of the protection instead. Even so, given that Lion's hull was roughly 600,000 cubic feet larger than South Dakota's, it seems reasonable that Lion's hull was heavier.

Another difference came in the armour thicknesses and layouts. Lion had much thicker armour than South Dakota, that covered a much larger area of her hull. Lion's belt armour was 15in thick, compared to 12.2in on South Dakota. Lion's deck armour was 6in over her magazines and 5in over her machinery spaces, while South Dakota had 5.3in over the magazines and 5in over her machinery. South Dakota's armoured deck also only covered her citadel, which contained the magazines and machinery. British practice was to extend the armour deck forward and aft of the citadel, protecting more of the ship against bomb damage and plunging shells. The British 5.25in turrets had more armour than the American 5in turrets. However, the USN gave their main battery turrets much heavier armour than the RN did; the South Dakotas had 18in turret faces, compared to 14.7in on the Lions. The South Dakotas also had a heavily armoured conning tower, while the British had a thinly armoured bridge structure. If we add together the weights for the protection and hull, we can get a fair comparison between the tonnage of these for the British and American design; Lion's hull and armour weighed 29,190 tons, while South Dakota's weighed 26,707. This accounts for ~2,000 tons of the difference in tonnage between the two designs, but there's still ~1,000 tons between their standard displacements.

This difference ultimately comes down to differences in equipment fits. The British chose to use merchanical hoists to move shells around the magazines, handling chambers and turrets, while the Americans used man-hauled shells. The American methods made for lighter turrets that were more resilient to damage to the ship. The British method was heavier, as it needed more equipment, but it was safer in bad weather (as the shell was always under mechanical control), and ensured a consistent rate of fire in long engagements, as it was much less tiring on the turret crew. The British electrical system was also heavier. They chose to stick with DC power, which made distribution systems and motors simpler, but required heavier wiring to carry a similar current. They also used heavy lead insulation, which further increased the weight of the system, compared to the plastic insulation on American cables. Finally, the British devoted more weight to their aviation facilities. The Lions had a heavy cross-deck catapult and hangar in the middle of the ship, which weighed about as much as a main-battery turret. The South Dakotas had no hangar, just two relatively light rotating catapults aft.