You've already hit on the biggest obstacle to using one: it's a priority target that is not particularly durable. More than most things it needs to use cover and terrain to avoid being the first head on the chopping block, and if you have the opportunity to use cheaper and less effective units as screens that's a good place to do it. That means that you make sure the screening unit is a Medium range with a TMM of +3 or worse and the Mad Cat III is at long range so that the other unit is the more attractive target so your opponent can either risk wasting shots on a poorer shot to take out the Mad Cat III, or they take the easier shot at the less valuable unit. Alternately, you can also screen with a heavily armored unit that's likewise worthy of attention. In either case, reducing line of sight from as many enemy units as possible is still paramount.
In this way, the Mad Cat III is best used as a striker/skirmisher, avoiding committing to the main fight until a target of opportunity (isolated by terrain, within one-shot-kill territory, or with a good position out of multiple firing arcs of enemy units available) presents itself and then take the shot, overheating if necessary, then fade again. Much like in Total Warfare, knowing when not to shoot is an important and difficult skill to learn. IF3 really helps with this, because you can more easily justify keeping a prime damage dealer out of the fight contributing indirectly.
The Mad Cat III in such an engagement can frequently end up facing opposing skirmishers, especially the ones fast enough to force a confrontation. In a pinch, the Mad Cat III also functions as a good scout hunter, because most of the things that can catch up to it are not going to survive more than one round of fire, and anything that ends up in Short range of it is probably going to just die to an 8 damage sledgehammer at low numbers, especially if you're supporting it with any smaller units that can contribute to finishing the target off.
