For larger sharks and ones with teeth, I give anesthesia and do it in the water bath. Anesthesia has its inherent risks, is expensive, and likely takes a toll on the liver, where it is metabolized. This is often required at the same time the shark is being treated with other medications that may effect the liver.
This female remains week and hardly resists the procedure, though she is getting stronger. In the water bath and awake, the sharks tend to fight and resist. This makes trauma from tube insertion and placement more likely. You also do not want to move the shark while the tube is in place. Also, some of these small, tide pool sharks can withstand extended periods out of water (hours) and low water oxygen saturations.
As for air, fish have swim bladders and are often eaten whole. Tiger sharks engulf air to become buoyant, so small amounts are easily tolerated.
This particular shark stopped eating during a very bad infection/infestation. I was able to identify three different paracites: amphipods, trematodes, and nematodes, but that is for a different post. She went anorexic before treatment, but I've seen meds cause it too.