SO, I took off from Wright Northern Isles in my Curtiss Sparrowhawk. Once airborne, I decide to spawn a second one, this one alighting from the Tiny. I used the Air-to-Air targeting to locate it, but it soon disappeared after I targeted it. Thinking the Sparrowhawk couldn't clear the Tiny's deck, I spawned on the Tiny and took off with no problems. So, I spawned another Sparrowhawk on the Tiny, but it too disappeared quickly. I then turned around so I was looking at the Tiny and spawned another Sparrowhawk. Turns out, they weren't spawning on the ship, but at the spot the ship was when itself was first spawned when I took off.
tl;dr: AI spawn point for the USS Tiny is stationary, which is bad when the ship in question is moving.