Arduino : distance measurement between fixed points
Seems to me you are looking for some kinda of SLAM thing. Depending on your knowledge in control theory and your precision goals you can go from simple rotary encoders on wheels to multiple sensor fusion like Extended Kalman filters (example videos). Probably the most precise thing you can have is sensor fusion with LIDAR (can be achieved precisions like 1cm), however the price range is way to far from described.

I would start with a small approach and go from there depending on the results I'd get:


rotary encoders (easy to calibrate and can be precise enough as a starting point)
9DOF IMU (use a Extended Kalman filter or similar to make an AHRS correction). And in this part I have many advices. You have suggested MPU-9150. I'm not familiar with it but looking at datasheet it seems very good, specially because gyro comes factory calibrated. In past I've used razor IMU / GY-85 (adlx345 + ITG3200 + HMC5883L) and gyro was very precise and low noise. Hope Invesense used the same sensor in MPU . First in hardware: Acc are easy to calibrate just use gravity. Gyros are easy to calibrate but hard to do. Why, because it may be hard to get something that spins at constant RPM and know that value. But as I've said, according to datasheet, MPU comes factory calibrated. Magnetometers calibration can be a bit tricky. Bottom line is you get readings from various points of your robot  in 3D space. you plot all and you will get an ellipsoid. Then you find a transformation equation that maps the ellipsoid into a perfect sphere take a look this article. Your concerns about Angle random walk can be minimized with kalman filter and similar. This page provides much great information and examples in how to make a good AHRS. Mr. Madgwick wrote a really nice algorithm that implements the IMU  filter and have it also written in C (nice for Arduino).


After integrate all things with Kalman and see the results, if things are not good enough, you can add more sensor and provide a better estimation for example: sonars, rangefinders, landmarks with image processing and as suggested also, RSSI and triangulation.

These links might help you in finding your way:


Introduction to Kalman Filters and SLAM
Indoor Location Tracking using Received Signal Strength Indicator