One of the prime sensors for global localisation on land is GPS. This is not an option underwater because electromagnetic waves get absorbed quickly.
There are however alternatives, which provide navigation information which is not so easily available on land.
- Large Baseline (LBL) - is a method based on sonar, which works very similar to GPS, just using acoustics instead. For this you need transponders that ping at known intervals. The robot can than calculate its position based on the run length intervals. Because the timings are much easier than with GPS, coming up with your own solution might be feasible.
- Ultra Short Baseline (USBL) - works the same as the LBL, just that the transponders are very close to each other (usually one device). In this way you get an angle and range measurement. Less accurate, but more compact.
- Doppler Velocity Log (DVL) - uses the doppler effect in multiple directions to give you the velocity against ground.
- Pressure Sensor - you can directly get a measurement of your depth in the water column with this.
- AHRS - based on IMU and Compass, you can get a good estimate of your orientation.
The bad news is that the sonar based methods are way off in terms of your budget. However, if you are interested in this, it might be feasible to construct a localisation method based on DIY hydrophones. Should definitely be a fun experience.
Your alternatives, especially feasible in a pool, is to go optical. You can for example use an external camera to track your robot, or use markers (e.g. Aruco) to get a relation to a known position.