Is this even possible - is there something like a RELEASE_VERSION environment variable?
There is now a released Heroku Labs feature called Dyno Metadata that gives you this information. Once you enable it, your running dyno's environment will contain environment variables with the Heroku version ID of your app, the git commit hash from which your release slug was built, and much more. For example:
HEROKU_APP_ID: 9daa2797-e49b-4624-932f-ec3f9688e3da
HEROKU_APP_NAME: example-app
HEROKU_DYNO_ID: 1vac4117-c29f-4312-521e-ba4d8638c1ac
HEROKU_RELEASE_VERSION: v42
HEROKU_SLUG_COMMIT: 2c3a0b24069af49b3de35b8e8c26765c1dba9ff0
HEROKU_SLUG_DESCRIPTION: Deploy 2c3a0b2