Is the semantic component of a generative grammar especially difficult to incorporate in psycholinguistic proccessing models?

You're asking two different questions between the title and the description.

1. Is generative grammar difficult to incorporate into cognitive processing models? Not particularly. In fact, because generative grammar constraints what is meant by language. This makes it much easier to design experiments within the paradigm. However, you have to remember that, as originally formulated, generative grammar does not make any predictions about the online processing of language. It is exclusively concerned with 'competence' rather than 'performance'. But it is easily laid out into experimentally isolatable components, and a lot of psycholinguists have chosen to adopt its general approach as a sort of 'basic linguistic theory'. Unfortunately, this has led to a significant impoverishment of psycholinguistic research in the last forty years or so.

2. Is the semantic component of generative grammar difficult to reconcile with psycholinguistic models? This makes the assumption that there is such as thing as a coherent semantic theory of generative grammar seeking cognitive plausibility. All the theories I've looked at are based on some version of formal logic. This is not too difficult to model experimentally but if you look at actual psycholinguistic research, nobody looks at this much (beyond some work in inference and presupposition). That's because natural language semantics is much more difficult to map onto generative linguistics than natural language syntax. So in practice, most psycholinguists rely on a sort of folk theory of meaning informed by various models.

But the question that would be more interesting to ask is: Do we have enough experimental psycholinguistic evidence to support a comprehensive processing model of natural language semantics and pragmatics?

The answer to that is definitely NOT because of point 1. We need a far richer experimental paradigm within psycholinguistics that is not confined to the lab (I find Discursive Psychology very inspiring in this respect). We need psycholinguists to look at the broader semantic research (and not just the scraps off the table of working syntacticians). We need to jettison primitive concepts based on the dictionary model of meaning and look at semantics as a process that happens both consciously and unconsciously, individually and collectively, cognitively and affectively. We need to once and for all bury the semantics/pragmatics distinction. We need a non-storage model of memory and recall. Far more research on framing and cross-linguistic research in non-WEIRD contexts.

Some of these things exist in germinal form but we're far from a model of how meaning is processed that could serve as general hermeneutic underpinning for the interpretation of a broad range of phenomena.