No, it's not.

When lifting the other character you are either taking an action, or you interacting with an object -- depending on your DM's rulings.

If the DM rules that the pile driver is an attack (which I'd argue it is), that would cost you an action, and you can only repeat it if you have movement and an action to spend on it. This assumes you've grappled them, and have an attack left.

If the DM rules you are use an object. If it wasn't an enemy, you'd only be allowed to pick it up once. Once you drop it (which is free), it is no longer a free action to pick it up again. Use An Object action:

You normally interact with an object while doing something else, such as when you draw a sword as part of an attack. When an object requires your action for its use, you take the Use an Object action. This action is also useful when you want to interact with more than one object on your turn. (PHB 193)

Since you can't lift someone and do something else, it might take a full action to lift someone above your head. Remember that a round is 6 seconds, dropping someone more than once (or twice if you have the actions for it) would take about that long. Moreover, really can't lift someone and jump while doing something else, so it could be argued it can't be an interaction and must be a "Use Object" action.

To rule otherwise would allow a free 5d6 damage without using any action, and that, while awesome the first time (and a DM may allow it once, by Rule of Cool), is game breaking.

Point of clarification: the question as you pose it in your title asks if gust can move "someone" under the effects of levitation, but in the body of your question you are specifically asking about moving yourself. I will focus on the latter.

By Rules as Written? No, absent DM fiat.

The general rule is that spells do only what they say they do, and anything more is up to the DM. The description of gust, which you've quoted, says it can be used to push others, but says nothing about moving the caster.

To put this in perspective, consider that you are effectively looking to use gust to target yourself on the receiving end of the push effect. Technically, targeting yourself might be fair game; the spell says you can target "[o]ne Medium or smaller creature that you choose," and you could perhaps choose yourself. The trouble is, the target is "pushed up to 5 feet away from you," and you cannot be pushed away from yourself. The DM could choose to ignore that bit of illogic, but it would be fiat.

By Rules as Fun? Absolutely yes.

Gust is an underwhelming cantrip. Most of its effects are duplicative of other (better) cantrips -- except for the push effect, which can be achieved just as easily by any character using the rules for shoving a creature. See PHB p. 195. Permitting a PC to cast gust in order to achieve 5 feet of lateral movement while levitating rewards creative play, and is highly unlikely to break the game. (Consider, for example, how difficult it would be to abuse this setup in combat. The PC is expending an entire action for 5 feet of movement.)