RF signals are attenuated greatly by water, Very low frequency (VLF) is used for submarine communications but requires a huge antenna system. Therefore Bluetooth and other RF-based systems won't be suitable.
For a depth of 30 meters accoustic modems are a practical solution. I've done some software programming for a system that used a AquaComm: Underwater wireless modem and it has an RS232 port and simple to use commands. From memory the board and transducer combined would have taken up an area of around 100 x 100 x 200mm, so I'm not sure if that would meet your definition of "small" or not.
Optical communications would be another possibility, although the only shipping commercial products I could find had a much shorter range. I found an interesting paper Using Optical Communication for Remote Underwater Robot Operation where they reported 30 meters over the length of a pool, although that dropped to around 9 meters in a harbor. With optical it would depend a lot on the water quality and ambient light.
Because it sounds like your device is always tethered by a rope the simplest and cheapest and easiest way might be to replace the rope with a cable. Assuming there aren't any dire consequences to the sensor failing and long-term reliability isn't an issue maybe standard mains cable would do the job if properly sealed at each end. It sounds like this setup it for some form of experiment?
As mentioned in a comment at that depth keeping your device watertight won't be easy so generally keeping the underwater component of the system as small as possible and removing the need for battery access will make it easier. For the same reason it might also be worth considering leaving the Arduino above water and just place the sensor(s) underwater if possible.