SO. . .
A pair of SeaHawks
16/32 cell VLS w. ESSM / ASROC
57mm or 76mm
RAM launcher
Several Mk 38 gun mounts
8 NSM cells
If we can find volume or are using a trimaran hullform which grants a wide hangar, I'd like to increase the helo complement to "a pair of Seahawks and a pair of VTOL UAVs", assuming the VTOL UAVs are roughly half the size of a Seahawk.
Even with quad-packing ESSMs, 16 cells is kinda painful. 6x ASROCs means only 40x ESSMs. Plus, I suspect that someone is going to insist on strike-length cells so that the US can continue its tradition of Tomahawk Diplomacy. Which suggests a minimum capacity of 40-48 VLS with 8-16 Tomahawks loaded. 6x ASROCs is IIRC only 2 engagements, so I'd honestly expect a dedicated ASW ship to have more like 12-16x ASROCs. But let's continue assuming only 6x ASROCs, so we have 24 cells remaining out of that 40-48. Now we can add SM2s or Patriots to the main battery and could load ~16x SM2s/Patriots plus 32x ESSMs (or however you care to balance that load).
Gun I would prefer the 76mm because of DART and VULCANO ammunition, but it is unlikely for the USN to adopt a new caliber at this point.
Agreed on RAM launcher, might go so far as to ask for multiple. 1 is required, 2 would be better.
On the Mk38s, I would want a radar on the mounts in addition to the EOTS on the Mod4s. And I want 4 guns for coverage.
And yes, 8x NSM/Harpoon/whatever light AShM sounds right.
Propulsion ?
Speed ?
Displacement?
Propulsion is absolutely going to be IEP, so it doesn't matter what the prime movers spinning those generators are. High end generation is almost certainly gas turbines, hotel load could be diesels or GTs. Needs to be properly rafted/silenced for ASW work, however!
Edit: Needs to have at least double the battery capacity of current ships, so that you can normally operate the ship with the battery between 25-75% and always have a little extra space so your running generators can be pushed a little harder to get onto their best-efficiency setting.
Speed,
if we're going to insist that this ship escort carriers, needs to be 30+ knots. Else we can accept ~25-28kts to stay with/ahead of merchant ships.
Displacement would be "however big the systems need it to be". If SPY6v3 needs 7000tons to be sufficiently stable then we have a 7000ton "ASW Frigate" and you tell Congress "that is how big the ship needs to be to keep the radar stable enough. Steel is cheap and air is free."
Judging on displacement jump from Spruance to Burke IIIs, we're likely looking at a minimum of 20% displacement growth over an FFG7. Which would suggest an absolute minimum size of about 5000tons with very little growth margin.
Displacement jump from Burke III to Zumwalt is 61%, which gets us to a minimum size of about 6600tons.
So, yeah, looks like 5500-6500tons is a not-unreasonable expectation for the FFG, and 7000 including extra growth margin.