Historically overweight weapons were used for training. The purpose was to make your sword arm stronger and more resistant to fatigue. If your arm gets too tired to fight effectively while the fight is still on you generally die.
The downside is that since overweight weapons have more inertia you do not learn to use the normal weight weapons with the speed and precision they are capable of. Because of this overweight weapons are not really used any more I think. Unless you expect the students to face a chaotic life or death melee of unknown duration training with the actual weapons simply works better.
Historically such combat was expected and it was often too chaotic and exhausting to use any "fancy stuff" anyway. Or more precisely after the fight got started exhaustion and confusion would handicap you anyway so training with overweight weapons wasn't a real problem.
Additionally more technical the skill, more teaching time per student would be required. When training an army it is better to train them to have strength and endurance with basic skill than any real technique and finesse. The economics work better.
But that is all about training, what about actual use in combat? Well, some exceptionally strong individuals did use overweight weapons in combat. Although it has always been rare since, like Gary Walker pointed out, it does usually compromise your ability to defend your self with the weapon.
So the first criteria should be that you do not need the weapon for defense. Heavy armor and shield (for one handed weapon) should be required. For your scenario, if everyone else is using guns, defensive aspects of swordplay might be irrelevant.
Second criteria should be that you can choose when to fight and when not to, so that you do not suffer from fatigue caused by the extra mass. A leader with body guard to watch his back, a guerilla fighting fast hit-and-run actions and so on.
Third, you have trained with the overweight weapon a lot and have little experience with normal weight weapons. Typically, this would be someone who wants to be strong and because of that trained obsessively with heavy weapons to get stronger. Actually learning to fight should have been an afterthought at best.
Fourth, there should be an actual benefit from using the overweight weapon. Usually this would have been psychological. If everybody knows you fight with a weapon too heavy for others, the effect on morale can outweigh anything you actually do with the weapon. You might be so strong that relying on your strength to overwhelm your opponents might be a reasonable tactic. In such case using a really heavy weapon might make sense since it would allow applying more strength to beat the opponents defences. You might simply not be confident in using anything other than your training weapon.
Negating gravity does not seem to be anywhere in the criteria for using overweight or oversized weapons. As others have mentioned it is generally the inertia that matters.
The kind of weapon would depend entirely on the reason an overweight weapon is used. A sword or axe of some sort is generally used for psychological effect. A staff or mace might work for applying strength. And of course, if the reason is because you do not know how to use anything else, that, whatever it is, is what you use.