In my approach, even in the rest frame of the first drone to accelerate, the distance to the next drone changes.

That's because 1) the drones do not exhibit Rindler motion, and 2) there are discreet gaps between them, not true of a mathematically solid object.

The second point can be resolved by simply noting that the proper separation between two objects is momentarily undefined since the neither drone is stationary in the rest frame of the other. It can also be resolved by making the object continuous instead of discreet.

The first point can be resolved by having all the dots accelerate per Rindler motion, which means each has a proper acceleration of 1/R where R is the distance to the Rindler horizon, which is the left dot if it has infinite proper acceleration. This is what yields the 5.51 day number.

Before the acceleration it is 1 unit, immediately after the acceleration, the distance is 2 units, and then over time, as the first drone approaches the second, the distance reduces back to 1 unit.

Agree. Proper separation is undefined during this short interval.

The changing distance would lead to "the string breaks," so my solution doesn't really work.

No it doesn't break because the string in continuous, not moving at all at first, and more and more of it moving (with instant acceleration) until the acceleration 'wave' reaches the next drone. This all assumes that the 'string' is self propelled (each bit of it), which of course just reduces the string to just another line of drones, which people tend to think of as discreet.

Note how the graph is similar to mine, but with hyperbolas instead of straight lines. This makes the length contraction exactly what it needs to be for the "string to not break" so-to-speak.

Yes. The page doesn't discuss reversing the procedure to bring the 'train' to a halt in S, but it's just a matter of starting from stationary in T and accelerating 0.0151c the other direction.

Note that in my example (with natural units of a light day), the acceleration of the right-most point is 1/365 light days/day² which is slightly below 1 g.

What really happens, (I think), is the following. In the rest frame of the first drone to accelerate, before the acceleration the distance to the second drone is 1 unit, immediately after the acceleration, the distance to the second drone is still 1, because the second drone had already launched (time difference due to RoS).

Let's compute it then. RoS computation: t' = Lv where L is 1 and v = 0.866. Here, t' is the time change at what you call the 'launchpad' of the 2nd drone. Pretty impressive for a point-size drone to get one of these.

Anyway, RoS time change at the pad is 0.866, and the drone did its acceleration at time 0.577, so indeed, the 2nd drone is already moving in that frame. All of them are as a matter of fact.

It still isn't Rindler (hyperbolic) motion, but what you say here works.

I tend to think the only true solution is the Rindler one.

Depends on what you call a 'true' solution. I did specify somewhat relaxed requirements in the OP precisely to allow a solution like the one you gave.

Did you compute how long it takes to move the light-year object this way?