We have managed to get attribute routing working happily from a module with K11.
Its not very well documented, but the steps are:
- Follow Kentico's documentation to start with
- Override global exception handling
- add /customapi to System \ URLs and SEO \ Excluded URLs
A good guide (credit to Sean Wright) is here:
https://www.wiredviews.com/blog/april-2016/integrating-kentico-cms-with-asp-net-web-api-2