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The basic lift requirements of a US amphibious force can be broken down to five key characteristics or fingerprints: embarked troops, vehicle stowage (square feet), cargo stowage (cubic feet), aircraft spots (CH-46 equivalents), and landing craft spots (LCAC equivalents). RANG has a useful study (here [1.6 MB PDF]) that breaks down these requirements for both ESGs and Marine Expeditionary Brigades (the MEB is actually the USN's main planning target--the goal is to have 2.5 MEBs worth lof lift in all five parameters.) For an ESG, the current targets are:

Troops: 2,924
Vehicles: 49,600 ft^2
Cargo: 172,700 ft^3
Aircraft: 47.4 spots
LCAC: 6.9 spots

The RAND study projects that by 225, those figures will ahve risen somewhat due to the crowing size of Marine equipment (expecially aircraft). The 2025 numbers are:

Troops: 2,925
Vehicles: 64,000 ft^2
Cargo: 197,000 ft^3
Aircraft: 53 spots
LCAC: 8.2 spots

Simple math says that for a 3-ship ESG based on a single ship design, each ship needs the following characteristics to support the future ESG:

Troops: ~1,000
Vehicles: ~21,400 ft^2
Cargo: ~65,700 ft^3
Aircraft: ~18 spots
LCAC: ~3 spots

A Wasp is too large for this: it has 45 VTOL aircraft spots, 145,000 ft^3 of cargo and almost 2200 embarked troops . To get an ideal single-design ship, you'd want to eliminate more than half the aviation capability, troops, and cargo space (LCACs are right on and vehicle square is a bit short, but close). The closest existing design I know to this is the Spanish BPE, but that's way short on LCAC space.

Personally, I'm skeptical that a single ship design would be at all the right approach, since the demands of aviation and landing craft operations are not necessarily the same (the dock wants to be closer to the shore than the flight deck, usually). My preference would probably be a two-design approach with each ESG being built of four ships (the better to support split-ESG operations): two aviation ships and two dock ships. I'd have to crunch some numbers to give you the right mix for each ship, but two distinct types is much more adaptable than a single design. The aviation ships would each have half the current force's troop lift aircraft (CH-46 or Osprey), with one taking the STOVL aircraft and the other taking the heavy-lift, attack, and utility helos. These ships would have small well-decks, big enough for one enlarged LCAC (1.5 spots), about 1000 troops, and somewhat smaller square/cube numbers than the current flight-deck amphibs.

The dock ships would need to be larger than the current LPD-17s, with 3 LCAC spots each, about 500 troops, and much enlarged cargo cube capacity. Aviation capability on the dock ships would be similar to the LPD-17; a couple of flight-deck spots but hangar space for only one or two aircraft (on loan from the aviation ships). Their flight decks would be used mainly as refuelling/rearming points for the attack and utility helos during a landing, but a hangar is desirable for independant ops or for hosting other aviation units (Naval Special Warfare helos, or Navy or Marine VTOL UAVs, for example).