I can relate your question/problem to my recent project what i have been working for (in fact i just put the similar question on the stack exchange but unfortunately did not get any answer from anyone).
let me just describe my project (you may be able to relate or find approach). So mine was not a mobile app or small desktop application but it was an integration of 6 individual big data application. The CEO found it potentially the next gen big data solution. Out of these 7 apps many have already its own customer base and have been adding millions to the company's revenue.
We received the business objective from the product owner, the VISION of the product were explained to us. Then there were lots and lots of meetings with the independent owners of the apps. You might imagine the complexity of the project. But as a UXer it is our responsibility to help product owner enabling them seeing the clear picture.
We finalize to use existing codebase (html, css and jquery) as much as we can to reduce UI development effort, So we designers were suggested to pick up similar behaviour from the apps and split them into template like Table Grids, Header-Footer, Navigation Interaction, Canvas, Login etc. We liked the suggestion and designed templates and re-coded (edited) them to make it ready to be deployed.
I just want to ask you if you really find problem in designing the app because the problem you defined is not the problem but it has potential business solution. I believe your firm wants to solve problem for their customers and give them better experience by simplifying the process (I guess that's what we UXer do). You may find examples around that how different solutions may merge as one and serve to the users as a unique solution http://www.networkworld.com/article/2848465/uc-voip/microsoft-rebrands-lync-as-skype-for-business.html
Conduct interview with product owners
Try to understand the problem/ opportunity (business need), If required do multiple meetings with prepared questionnaire to solidify your understanding of the business goal.
Analyse both the products
with UI developers that whether you can use existing codebase for new UI which you will be proposing.
If the developers feel that codebase will not require much of the re-do work then i presume you are good to go in designing single solution. Else convey the owner that it requires (70% or 50%) new codes.
Estimate your design and development accordingly and ask if they aligned with it, if they don't agree you just surrender (Because you must support best design and should not throw any garbage into software industry with poor usability).
I believe it should help you!