Van Fraassen’s own argument for Constructive Empiricism is minimal. He sketches out both Scientific Realism and Constructive Empiricism and then argues that there is no reason to prefer the former to the latter. But you might not agree that the onus of proof lies like that and want a more explicit argument. Here is what is implicit in his thinking:
At any given time, the only evidence for a theory is the set of relevant observations already made.
For any finite body of observations, there is always more than one theory that is consistent with it. These different theories differ in what they say about unobservables. (They might also differ in what they say about future observables, but we will ignore this point for the moment.)
As the only evidence for theories concerns the observable realm, there is no way of deciding which theory correctly describes the unobservable realm.
Thus we should not take the risk of choosing between these different theories and accept instead what they all have in common: their descriptions of the observable realm. We should, in other words, adopt Constructive Empiricism.
However, as all the consequences that can matter for us are observable we also loose nothing by this.
This argument is a form of scepticism about unobservables combined with a prescription for what to do in response. Since we cannot know about unobservables we are best off not taking the risk of making claims about them.
The argument from combining theories
If one takes a Constructive Empiricist attitude to a theory then one should believe only its observable consequences. Thus if one adopts this attitude to two theories one should believe the combination of their observable consequences.
But the observable consequences of the combination of two theories is not necessarily the same as the combination of the observable consequences of the two theories considered in isolation. The reason is this. The unobservable (or theoretical) component of one theory may ‘interact’ with either the theoretical or observable component of the other to generate fresh observable consequences, i.e. consequences that flow from neither theory working in isolation from the other.
Given this fact, it seems that Constructive Empiricism is an inadequate philosophy of science because it cannot account for the fresh predictions that result from combining theories.
Sadly, however, the combination of theories does nothing to change the basic position. The observational consequences of their combination still underdetermines the microscopic (unobservable) realm. Thus there is still no reason to believe in anything more than the empirical adequacy of the theories in question whether we consider them working together in combination or separately in isolation.