I think you need to check your assumptions or recheck what you think you see, because what you are describing seems almost impossible, unless your kids are driving you crazy by scooping cups of water out of the tank while you're running your tests.
Critically, you say that dye in the tank does not make it into the bowl, but that the water lost from the tank is not ending up on the floor or between the pieces of toilet. Regardless, if it leaves the tank, it has only three places it can go: (1) back into the water supply (a siphon that anti-siphon valves are designed to prevent); (2) into the bowl -- either around the flapper, down the overflow tube, or through a crack or defect in the flush valve or its seal; or (3) onto the floor or otherwise outside of the tank into the room. Taken together, your premises seem to foreclose that, which is odd right there.
You do say that if the water is leaking past the flapper the whole tank would drain. That's not necessarily true; water pressure above a certain level could result in a leak of only enough water to get the pressure down to where it isn't leaking. However, I think that is probably less likely, because you say that dye isn't getting into the tank. Otherwise, I would say check the seat in the fill valve and make sure it's smooth; that's why Korky makes that kit that contains silicone, a repair ring for the valve seat, and a new flapper, and there are plenty of reviews that say, "I still had a leak after changing the flapper, but then found this kit and it solved my problem."
The most obvious candidate, because your toilet drains to the same level each time and then stops, is a siphon through the fill tube. You say it isn't siphoning, but you didn't say, "The refill tube is positioned on the end of the little clip and perched a quarter-inch above the lip of the overflow riser." So I would just want to make sure that that is your setup and that you don't have the fill tube stuck down in the overflow riser, regardless of whether you think it's siphoning. That could also account for a reduction in tank volume without water turning the dye color; the siphon would first pull undyed water. However, it wouldn't pull that much undyed water, so if you put a dye tab in the tank and then it drained 1-1/4" (which is likely more than a quart) and dyed water did not end up in the bowl, back into the supply, or onto the floor, it's a mystery.
Perhaps a crack or leak in the overflow riser at the location that the water drains to? Still, water down the overflow riser ends up in the bowl, so you would see that with the dye.
How is your water pressure? Is it normal? What kind of fill valve do you have?
I will be interested to see what the real experts on the forum come up with, although I have to say that, respectfully, it seems like one of your assumptions isn't right.