How do you predict a cognitive bias theoretically using dual process theory?
Well, whether something is truly predicted or a post-hoc explanation can be difficult to disentangle in this area. What you're asking for is that dual-process theory predicts the existence of a new type of bias previously unobserved. And that's a tall order because observational psychology has been around for a log time. 

Probably the area in which I buy the dual-process explanations the most is belief bias; dual-process theory predicts that our intuitions (T-1 processes) bias us for plausible conclusions (and against implausible ones), even if the reasoning (T-2 process) is faulty (and respectively correct).

Predicting an entirely new type of bias is not the only way to test dual-process theory. One can test it quantitatively by looking at known biases and verifying experimentally if the timing affects the level of bias. And in the case of belief bias, that seems to be the case.

The purpose of any scientific theory is to explain some aspect of the real world. Just because we don't continuously discover (qualitatively) new facts based on that theory doesn't invalidate it as scientific theory.

In psychology in general, experimental data is fairly easy to come by, so evidence often leads theory. It's not like in modern physics where you predict the Higgs boson and it takes decades to discover it in actual experiment. Or you predict the string structure of the universe, but it's unlikely we'll ever be able to test that.



By request, here's the introductory / "theoretical" part of the paper:


  [Dual-process] theories posit two distinct
  processes of reasoning that compete for control of the response that
  participants make in reasoning tasks. Heuristic or System1 processes are
  characterised as rapid, implicit, associative, and heavily contextualised,
  whereas analytic or System2 processes are described as slow and sequential
  but capable of abstraction and generalisation. Note that a key difference is
  the speed of processing. The reason for this is that analytic processing is a
  sequential process requiring use of central working memory and is
  constrained by its limited capacity. By contrast, heuristic processes operate
  through massively parallel implicit systems that exert an unconscious
  influence on responding. In support of this distinction is evidence of
  substantial correlations between general intelligence and working memory
  capacity with abstract deductive reasoning but not pragmatic, belief-based
  reasoning (Stanovich, 2004).
  
  While a mass of evidence has been recorded supporting the idea that there
  are two different processes of reasoning, more research is needed to confirm
  the characteristics that theorists have attributed to the cognitive systems that
  may underlie these observations. A relevant methodological innovation
  reported by Roberts and Newton (2002) is the rapid-response reasoning
  task. The idea is that by constraining participants to respond within a short
  period of time, the slower analytic process of reasoning will be differentially
  inhibited. For example, on the Wason selection task, there is a well established
  non-logical tendency, known as ‘‘matching bias’’ (Evans, 1998),
  to select cards that match the explicit content of conditional statements,
  regardless of the presence of negations. Roberts and Newton (2002) showed
  that measures of matching bias were significantly increased when a rapid response
  version of the selection task was compared with a free-time version.
  This is as dual-process theory would predict, since any influence of analytic
  reasoning to inhibit the bias would be suppressed by the requirement to
  respond quickly.
  
  A central phenomenon in dual-process accounts of reasoning is that of
  ‘‘belief bias’’—the tendency to evaluate the validity of an argument on
  the basis of whether or not one agrees with the conclusion, rather than
  on whether or not it follows logically from the premises. The
  phenomenon is one of the earliest reported in the psychology of
  reasoning (Wilkins, 1928) but the modern study of the effect dates from
  the paper of Evans, Barston, and Pollard (1983), which established the
  phenomenon with all relevant experimental controls. Evans et al. showed
  that participants’ evaluations of conclusions were substantially affected
  both by the logical validity of the arguments and by the believability of
  the conclusions. With the help of protocol analyses Evans et al.
  characterised the effect as involving a within-participant conflict between
  logic-based (analytic) processes and belief-based (heuristic) processes.
  Such a conflict is to be expected in view of contemporary dual-process
  theories. In support of this, it has been shown that participants high in
  general intelligence are more able to resist belief biases (Stanovich &amp;
  West, 1997) and that logical and belief-based responding are neurologically
  differentiated (Goel &amp; Dolan, 2003).
  
  In this study, we extend the rapid-response methodology of Roberts and
  Newton (2002) to the study of belief bias in syllogistic reasoning. Using a
  conclusion-evaluation paradigm with problems similar to those of Evans
  et al. (1983), we compare performance under rapid-response and free-time
  tasks. We predict that the participants required to respond rapidly will show
  (a) an increased level of belief bias and (b) a reduced level of logical
  responding.




And regarding prospect theory; quoting from Wikipedia page on it:


  Critics from the field of psychology argued that even if Prospect Theory arose as a descriptive model, it offers no psychological explanations for the processes stated in it.


So I don't see how it is any less of a reverse-fitting of (arguably more math-oriented) description to prior data.

Furthermore, the endowment effect is not even certain to exist, if wikipedia is correct (There are some failed replications, no meta-analysis that I can find.) There are a lot of other (possible) theoretical explanations given for it there as well. In fact there is more theory than there are experiments. Also, being much closer to economics, it's generally more likely to have mathematical models for it.