This “problem” is entirely flawed from the beginning.
First, however, a note. Special relativity doesn’t simply say that there is no absolute reference frame. It assumes that changing the reference frame one uses does not change the fundamentals of physics and that the speed of light is the same in all reference frames. This means that for a large range of reference frames, we can choose whatever we want and we will get the same events happening according to the laws of physics. It also means that there is no absolute reference frame that is picked out by the events in physics; i.e., there is not one special frame where everything works out or where all the laws of physics have their true form.
OK, so now to the problems.
One thing that is a prerequisite to any discussion of physics is a reference frame for measurement. In non relativistic scenarios we can assume a vague reference frame, but in relativistic scenarios we need to understand what reference frame we are using because it matters to the calculations. In this case, we are not told the reference frames in which the “2 light-seconds (2Ls) apart” is given.
In this case, I could presume that both ships are at rest relative to one another and that the distance given is in a reference frame currently at rest with respect to the ships. However, this is information that should be supplied at the start.
We have to know what a “travel clock” is. Does this measure the travel against a given reference frame? The reference frame that should have been given above but wasn’t? And what was the speed relative to the given reference frame? Is it the measurement of how long the engine was firing?
The distance between each ship is not a constant thing, it is something that depends on the reference frame that we use to determine the distance. Through choice of reference frame, we can make the distance between the ships arbitrarily small.
I fear that in the above question there is a confusion between an absolute reference frame and a reference frame as an intellectual artifact. An absolute reference frame is a particular frame that is picked out by peculiar physical laws or phenomena; this is something that is effectively ruled out by special relativity. A reference frame is an intellectual artifact that we use to describe physical situations and always has been; this is something used by special relativity. In order to describe distances and use physics, we need a reference frame.
I am afraid that I simple do not understand what this is supposed to mean.