Adding SSRS content types seems to be a bit flaky.. We couldn't get the content types automatically added by activating the feature in one web application – so we manually added the content types into the site collection as specified in here (http://sharepoint.stackexchange.com/questions/111619/sharepoint-2013-sp1-installation-issue-ssrs-content-type-missing-for-new-sc) – this post also had some other good suggestions.
So we add another web app for another purpose – again same issue with the content types not adding, so we add them manually again and get the following error when trying to create an item in the document library for "Report Data Source", "Report Builder Report" and "Report Builder Model" – 'New Document' requires a Microsoft SharePoint Foundation-compatible application and web browser. To add a document to this document library, click the "Upload Document" button."
We had added them manually correctly and obviously it was installed correctly as another web app was working..
We came across this post which worked for us! https://chrishattonnzlsp.wordpress.com/2015/07/21/reporting-services-new-document-requires-a-microsoft-sharepoint-foundation-compatible-application-and-web-browser/
Running rsSharePoint.msi fixed the issue. We had to delete the content types from lists and the site collection and then deactivated and reactivated the reporting services feature, the content types were there this time and we could add the proper ones into the libraries.