Unfortunately, it doesn't look like we have an API to obtain this at this point in time.
I checked the /api/version end-point, but it only returns the "2016.3.6.0" portion of the path...the rest has to do with build #'s and cache buster identifiers.
For all intents and purposes, you should be able to manage a configuration of the "current jive.css" path ... and hold a copy of the value. It should remain around and be fairly resilient. If you checked it once a quarter, or after your cloud update .... that would be the easiest way to keep it in sync.
I cant think of another way for you to manage this dynamically, but FWIW, the jive.css that impacts the internal representation of a document does not change all that often...so you should be fairly insulated.
Hope that helps.
Too bad an endpoint like that doesn't exist, but certainly agree that managing a copy of the current path and reviewing periodically is the best work around. Thanks for the insight around insulation.
Have a great weekend,