I'm currently working on a project that uses the Ti CC2650 BLE chip, and there is a sensor tag available for $29 that has incorporated a 9 axis accelerometer/gyroscope, you connect via blutooth to your mobile phone and you can read the various sensors, light level, temperature, humidity, pressure, magnetic sensor, x,y,z accelerometers, x,y,z gyroscopes, x,y,x magnetometers and microphone, runs for a year on a lithium coin cell.
http://www.ti.com/ww/en/wireless_connectivity/sensortag2015/index.htmlAndroid and IOS code is free download.
I doubt that the gyro specs are good enough for this application, refer to the data for the 9 axis chip MPU9250
http://store.invensense.com/datasheets/invensense/MPU9250REV1.0.pdfThe first problem, I see is that the gyro's will drift with temperature, so the experiment will need to be conducted in a temperature controlled environment, second is the noise in the gyro output, so some external software to average the gyro outputs over time. The data is 16 bit degrees/sec, and we are looking for something less than 15 degrees per hour, or 0.00417 deg/s, while the noise is 0.1 deg/s rms.
With a bit of careful temperature control ( and correction ) together with some clever averaging, it just might work, but I think it will be borderline.