Not using the TG5 but the TG4, so this might totally irrelevant, but I have had similar experience there. However, my understanding is that you are talking about the jpg version of your shots (as I am doing).
Using the raw data, you can indeed adjust the WB as you see fit (I suppose that that is what you are doing above), provided obviously that you have enough signal. I don't think you can expect a preset WB to get it right in all circumstances. In very diverse environment, you should expect to WB each shot specifically. I personally don't bother, as I am using my camera mostly for memory snapshots, but that's in principle feasible.
This being said, I am using video lights as well (no strobes). If close enough from my subject (I have 15,000 lumens of light), the UW mode does indeed seem to not get it right. I am too lazy to switch back and forth between mode, so I just use the raw version.
Another issue you might have is that you are apparently using your zoom lens (10 m, 15 m for the critter shots). I don't think you should expect anything vivid at these distances. Too much water between the sensor and your subject. Your lights are powerless at these distances. Get closer, dim the lights if that frightens the critters, but reduce the color absorption by water.