Christodoulou does not use Rendall's result to "set up his initial data". What happened in the end of Chapter 2 is that he appealed to Rendall's result to show that, given the initial data he has prescribed on the two initial intersecting cones, there is a local solution (in harmonic gauge). Given that a solution does exist, his arguments (the various estimates he proves in the middle part of the paper) guarantees the existence of a double null foliation along which the required boundedness of various fluxes hold. (In other words, once you have a solution, you can start proving a priori estimates, even in the situation when the coordinates relative to which you are trying to prove estimates are dynamic; this is the standard bootstrapping procedure in quasilinear PDEs.)
To get a solution that exists for long enough time, it is not sufficient to appeal to simply Rendall's result. So Christodoulou proves an extension theorem in Chapter 16: the proof spans page 559 - 580 on the arXiv version. The local extension theorem works with space-like initial data in harmonic coordinates. The reason that you can use space-like initial data is that, remembering that we are only interested in a region near the incoming cone, and within the incoming cone by domain of dependence arguments we know the metric is flat, as soon as the initial region of existence (which is guaranteed by Rendall's result) is found, one can draw a space-like hypersurface, which will, in principle, lie in the past of the first trapped surface that is to be shown.
(This is where the initial assumption of having a flat causal diamond is very useful. In principle you can also replace the Minkowski causal diamond with a causal diamond from any suitable known regular solution [for example a diamond from any of the Christodoulou-Klainerman small data solutions]. In terms of what Rendall did, the idea is that for Rendall's proof to work he needs to "fill in" the region to the past of his cone with "stuff" which he then subtracts as an inhomogeneity when solving the space-like initial value problem. In Christodoulou's case, he started in a situation where the "stuff" in the past of the cone is already known [in fact Minkowski], so he does not need to go back and fill it in.)
The extension theorem is a bit long to prove in part because he needs to translate the estimates he has proven in the double null foliation to estimates that would be useful to control the evolution in harmonic coordinates.
See J. Luk, On the Local Existence for the Characteristic Initial Value Problem in General Relativity for a treatment of related questions.