This might actually be way more challenging that you/your client initially expected.
SharePoint's UI is quite complex and it requires a deep understanding to enhance with custom functionality. The incorporation of a framework such as Bootstrap is certainly possible (unlike you were told, I've done it), but not a trivial task.
The typical scenario for a site that will use such a framework (possibly) incorporate responsiveness to support cross-device would be to engage the customization of a SharePoint Master Page.
There are several ways to do this, including SharePoint Designer, Visual Studio or even manually editing in a file and uploading to the MasterPage Catalog.
SharePoint Master Pages do require A LOT of care when applying customizations. Its structure cannot be freely modified as it would destabilize the CMS capabilities (ribbon, suite bar, Web Part editing, etc.).
For this reason, you will probably require a SharePoint Branding expert if the intention is to brand SharePoint itself.
The other alternative is to use a Provider Hosted App. Such Apps are hosted OUTSIDE of SharePoint and can be developed in virtually any technology, such as ASP.NET, ASP.NET MVC but including others as well.
The experience would be that the user will hit the SharePoint environment, and from there they can navigate to the "App", which will be what you are developing right now.
So, depending on the requirement it seems that you either:
a. Need to engage a SharePoint branding effort of Master Pages and Page Layouts
b. Need to research on provider hosted Apps for SharePoint 2013