export search_api settings with features
I was having this problem today too.  Features would not display my search_api index for export.  It also wasn't displaying my Page Manager pages, Page Manager handlers, or Views.  

I realized the problem was that I had moved all of those configurations into a custom module.  Features wouldn't try to export anything that is module-provided, even if those items had been created through that site's UI in the first place.  

I disabled the module (without un-installing) and Features was happy to let me export my index configuration after that.