Japanese got 600m height out of an AA Mortar with a muzzle velocity of 80-150m/s, don't have a proper source just MV of the shell, just used MV of standard mortars used to launch it. US 81mm mortars could hit 210m/s, 107mm 250m/s, so I'd say 105mm and higher necessary for 1000m plus, could be wrong and could do it with less. But still expect 105mm plus
Issue with a 24" mortar is the size. There was a 24" mortar deployed in WWII on a self propelled chassis, that was 124 tons. Presumably a naval mount would weigh similar when all was said and done with navalising it and handling recoil. Issue is it weighs as much as a pair of twin 5" DP mounts, and I'd consider the longer range of the twin 5" much more useful. I'd say AA mortar mounts would have to be small enough not to displace a Bofors Twin or larger weapon, if they were to be practical
Mind you I am of the opinion this would probably not work
The 124 tons seems a bit excessive, so lets look at a few things.
When I was in the Berlin Brigade, our companies mortar platoon had, IIRC, two different guns, one was (I think) the 81mm, while the other was (I think) the 4.2" and that is/is not the 107mm? I know that both were man portable, so let us do a quick mental exercise here.
Take the notional 24" and 36" AAA Mortars I have been on about, and let us look at space/weight. Now, a single 24" mount is going to take up a certain amount of deck space, and if we were to posit a similar space being taken up by an array of roughly 4" mortars, we could probably get away with three concentric circles of such tubes, from deck space taken up perspective, and this notional array could have 6 tubes in the inner ring, 12 tubes in the center ring, and 18 tubes in the outer ring, for a total of 36 tubes/mount. Of course, this isn't going to actually work out this way, but just to give us an idea, let's go with this for now. Ok, so if each 107mm tube weighs 100 lbs, then the total weight of tubes alone is going to be 3600 lbs, and this then gives us 1.8 tons as the starting place for our mount. Let's say that we take the weight of the tube as being a total of 10% of the actual mounts total weight, and thus come up with 18 tons total weight for an alternative to the 24" mount.
For the notional 36" mount, let's say that gives us a mount with 4 concentric circles, with 6-12-18-24 tubes in each ring, respectively. This would then give us 60 tubes for the 36" alternative mount, and this would then give us 60 100 lbs tubes, for 6,000 lbs/3 tons, and then a total mount weight of some 30 tons or so.
So now, let's amend the OP, to posit either a system of massive mortars in the AAA air defense role, or, alternatively, a multi-tube array system of AAA mortars in the air defense role. the single big-barrel mortars would probably be far less of a hassle to load and operate, but the munitions might prove to be a problem, while the massed array of smaller mortars would be a nightmare to load and operate, could the simplicity of their design prove to be the better of the two systems?
I'm a total novice in the realm of alternative air defense for 1930-1945 warships, so I have to speculate without any factual basis to really model things on, but am attempting to do the best I can, in the hopes of getting enough of a discussion going on that folks more knowledgeable than I will join in and steer us in the right direction.
That being said, could we see an interesting, and perhaps viable additional AAA anti-aircraft defense capability coming into service in the early years of WWII, based upon massed mortar fire being used to fill an incoming aircrafts attack profile with huge numbers of explosive projectiles? A notional DD, with ten of the multi-tube arrays (36 barrel type), could put some 360 mortar rounds in the path of an incoming aircraft, per volley, while the same DD, with the (60 tube array), could put some 600 mortar rounds out, per volley. Rate of fire would be an issue, of course, but my main concern is training gunners not to shoot at an incoming aircraft at all, but rather the volume of space their attack profile should have them transiting through, and then filling said volume with massed fragmentation projectiles.