I will present my thoughts rather than giving a definite answer in the Wolfram language, focused solely on configurator technology/methodology, and will give some humble suggestions. I have NO knowledge about ASP (Answer Set Programming) and in addition have absolutely NO knowledge about the scope and organization of your project.
I’m not affiliated with one or more of the mentioned vendors; I’m a mere user of one of them.
Configurators are a software approach to do mass-customization of products and services and are historically an off-spring of AI (artificial intelligence) research and expert systems applications. It needed almost a decade to from the mid 80´ies to the mid- to late 90’ies to make it “mainstream”. However configurator vendors and its users (i.e. developers, who do integrate the framework from vendors, and the modelers, who develop the models) cannot deny that it is still a very special field of application. Configuration as part of the business process needs to consider more things than the mere configuration task, but let's focus on the configuration task. For a little bit of introductory background I suggest , where a lot of pages can be read on Google Books. If you like to go deeper I found  for the classical TMS and  for a link to your desired info on ASP.
Evaluate commercial configurator solutions (ConfigIT, SAP IPC, ILog, Tacton, Selectica, CAMOS, …) - if not done already - to get a real sense about its power, its weaknesses, its maintainability, about it’s expressiveness on modeling, collaboration capabilities etc. Maybe ASP can be achieved or is somehow self-contained as a feature in those products. An awful lot of money went into the research and optimization of configurators of the big players. ASP is mentioned in research literature for conferences on configuration . Although very theoretical, you may give it a read.
If there is already a configurator in your business: do a prototyping on a fragment of the actual model in Mathematica with the Wolfram Language. Pick a particular constraint on a limited set of components and try to formulate it via
Cases etc. in Mathematica. Literature , ,  may help.
I would (from experience) go for OO-Methodology for putting together your product structure (component modeling) and PAIR it with with Constraint-based Reasoning Techniques to formulate your knowledge. As far as OO-Methodology goes you need at least simple construct for Part-of-Relationship and Is-a Relationship (Specialization). There will almost also be needed TABULAR relationships from which you can infer order numbers (part names) or do domain restrictions (on characteristics of a certain part instance). These tabular relationships (e.g. variant tables) represent knowledge which have been formulated before-hand by R&D or Product Management.
Other things to consider:
A) Interfaces to other Business process steps:
- Who and how many people are tasked to the modeling?
- How are the users of your potential solution? Sales guys in the field or internal specialists?
- What happens with the outcome of your configuration? Will it passed to ERP-System for sales order or furthermore BOM-explosion or Routing.
B) Performance, Performance, Performance
- Normally, a reasonable management tolerates the trade-offs of the outcome of engineering task like a SW package. But you never can talk about performance, reasonably, even if the sw runs correctly.
- What happens with configurators and big models (no matter if ill-defined or well-defined) is that at runtime the Pattern Matching System (PMS) might spend more time worrying (i.e. exclude invalid solutions from solution space) than actually give a quick valid answer set. That would be a sign of a needed partitioning of you configuration which might not be achieved within the framework of that configurator product but need to be developed by yourself.
To build a configuration solution exclusively in Mathematica is an interesting and compelling idea, but will it fly in brutal daily business? Is a compelling full-fledged-solution, e.g. in CDF, in a business context doable? Unfortunately, at the moment, I doubt it, it is a barren field as you stated in your question. Others may disagree with me, though, and I like to be proven wrong.
1) If you only need Mathematica to feed your model or map certain aspects of business logic into your model of an existent configurator or you like to verify aspects of your model you can certainly start writing
Assertion or do complex calculations with
NSolve for your domain. Automate the process so that the model or test results for the configurator can be generated.
2) If you intend to build a SOLUTION for your configuration problem with Mathematica and deploy with CDF or WebMathematica or Cloud prepare to take CONSIDERABLE TIME for an investigation and benchmark against what’s on the market already.
 Wolfram, Stephen. “Mathematica Book Version 1”, Chapter 4 (about Methodologies – this valuable chapter has been lost in later versions!!), Addison-Wesley 1988
 Wagner, David B. “Power Programming with Mathematica”, Chapter on Implementing your own DataType, Mc Graw-Hill 1996
 Maeder, Roman. “The Mathematica Programmer: Logic Programming I: The Interpreter”, Wolfram library archive http://library.wolfram.com/infocenter/Articles/898/
 Alexander Felfernig, Lothar Hotz, Claire Bagley, Juha Tiihonen. “Knowledge-based Configuration: From Research to Business Cases”, 2014
 de Kleer, Johan. “An assumption–based truth maintenance system”, Artificial Intelligence, 28, 127–162, (1986).
 Alexander Felfernig, Cipriano Forza, and Albert Haag. “Proceedings of the 16th International Configuration Workshop”. 2014