Potential energy of air inside the bubble definitely increases by mgh when the bubble rises by a height h. m is the mass of air inside the balloon.
When the bubble rises above, the volume it is occupying at lower altitude will be now occupied by water from higher altitude. Let us assume that volume V of the bubble remains same. Then decrease in PE of water will be M g h, where M is the mass of water of volume V. Surely M g h is more than m g h.
Explanation is that there is mechanical work spent in bringing the bubble up and moving water down. There is fluid resistance due to collision of particles. So there is a loss of energy there.
Further there is an elastic potential energy called Surface Tension. There is some energy stored on the surface of the bubble equal to surface tension multiplied by surface area of the bubble. If the bubble size increases due to less external pressure at higher altitude, then there is more energy stored on the surface.