In this instance, I respectfully disagree with @Wrzlprmft's comment. The core of this question is clearly focused on the research design / methodology and, thus, fits the scope of Academia.SE very well. Software development in this case simply represents the context of the applied research.
In regard to the essence of the question, I can offer the following insights and recommendations.
In my opinion, grounded theory (GT) is not a good fit for your study, as GT is typically used for designing general theories, rather than applied ones.
Design science seems like a pretty good fit, so you might read up and consider strategies for compressing research, performed, using this approach, to fit your time frame.
You might also consider other qualitative research designs and approaches, such as content analysis, narrative analysis and others (see more details, for example, at Qualitative Research Guidelines Project).
I would especially recommend you to pay attention to action research, as this research approach seems to fit quite well with your planned study. For more details on action research and other qualitative approaches, as well as likely the most popular qualitative research software ATLAS.ti, see this page.
Finally, I would urge you to pay attention to your terminology, as it might be very confusing not only to potential readers, but to the authors themselves. For example, your phrase "theoretical design of the software application" sounds... quite strange and reduces the clarity of your work and, thus, potentially, can hurt readers' impressions from it. I think that "conceptual design of a software application" sounds and reads much clearer. Hope this helps. Good luck with your thesis!