What isn't 100% clear is why a field that isn't "prefilterable" in the SOSL where clause would also be excluded from any SOSL indexes.
There is also the associated Idea : Include pick list values in search, which confirms other users finding that picklists are excluded from SOSL results.
Best Answer
Yes you can do SOSL query using REST Api.
The api would be -
/services/data/v37.0/search/?q=SOSL search string
The SOSL search string needs to be URL encoded.
Here is a link for more details - SOSL Search using REST API
Here is an way how you can make a successful REST API call.