The IJN managed to create a twin 5.5" LA that weighed 49 tons, the RN's 4" HA Twin weighed 16 tons. The lightest version of the Mark I 5.25" DP was 77.5 tons, and had extra weight added once in service (With RP, the one on Vanguard, it went up to 95 tons). The Mark II was in the 84-96 ton range. Realistically you could install a twin 5.5" LA and two 4" HA twin mounts for each 5.25" mount, and still probably save weight.
Dual purpose mounts often seem to come across as a false economy when you take a careful look at them. Its one of those things that seems like a good idea, but may not actually be so. The RN's DP mount that actually worked the 4.5/45 Marks I-IV ranged between a twin mount of 37 to 49 tons. For a carrier, and this is a hindsight thing, an anti-surface capability is an optional luxury. For the same weight British carriers could have had 2 or 3 times the number of 4" mounts, if an Aircraft Carrier has to depend on its own integral armament to fend off a surface attack then something has gone horribly wrong and its not going to matter anyway.
It is not necessarily an advantage for lighter vessels either, Anthony Williams makes a convincing argument that RN Destroyers of WW2 would have been better off using an all 4" HA/LA armament as well http://www.quarryhs.co.uk/MCGWW2.html in that it would of produced a higher rate and volume of fire.
As the Royal Navy already had a 5.5" gun in service, and it knows that a practical twin mount can be built from the IJN's example, considerable time and money can be saved in not developing the 5.25" gun and mountings.
(And probably the proliferation of other designs as well)
Good points, but as jsb points out looking at pure tonnage is all well and good, but tonnage doesn't' necessarily mean space.Weight wise you need to add the full ship impact, such as crew, deck size & length, fire control directors, magazines and shell handling areas..... three mounts means far more of each of the above even a twin 5.5" and a twin 4" will cost more weight once you add them in, especially as the Twin 4" HA RN gun is hand worked open shield type with out the shell handing and hoists included in the 16t.....
Deck space has to be taken into account.
In the early war I' put the 4.5s over the 5.25s in almost any take. Arguably with the fact they worked throughout the war better than the 5.25s gives them the advantage.
However by late war the bigger, larger caliber weapons having the improved fire control and the kinks removed became excellent weapons.
Still, look at Scylla. "The Toothless Terror"
You'd stuck your cheap bog standard 4" HA twin mounts and rigged it with a full fire control system made for a truly terrifying AA ship.