Does the combination work as one would expect?
Yes. In your case, the primary benefit of invisibility is:
Attack rolls against the creature have disadvantage
The cloak of flies will not impede this.
Another benefit of being invisible is usually that:
For the purpose of hiding, the creature is heavily obscured.
The reason that being heavily obscured helps in Hiding, is that
You can't hide from a creature that can see you clearly
and being heavily obscured prevents you from being seen, so that you can make your Hide attempt even while you are in line of sight.
What happens if you use the Hide action?
You can certainly Hide, but you are a surrounded by a cloud of flies which isn't Hidden. You are neither wearing nor carrying these flies, so they are still quite visible1.
What are the advantages of Hiding?
Make a Dexterity (Stealth) check when you attempt to conceal yourself from enemies, slink past guards, slip away without being noticed, or sneak up on someone without being seen or heard.
You are surrounded by a cloud of flies fifteen feet wide - are you realistically going to be able to do any of these things? About the only thing you might be able to do while hidden is move without being heard1, even though your position will still be noted by anyone who can see. In your case, your attempt to Hide could be used to resolve your attempt to move silently when surrounded by the obvious cloud of flies.
If you are unseen, and your attempt at Hiding means you are not making noise, it is possible that your attackers will have to guess your location - and if they guess wrong, their attacks will automatically miss. Tempting as that is, it quite depends on the intelligence of your opponents and how much prior experience they have had with magic.
Unintelligent opponents will likely treat the cloud of flies as a natural cloud of flies, without suspecting that you are even inside (although they might smell you, and conclude that you are in fact what the flies are being attracted to). Intelligent opponents with little experience of magic will likely recognize the perfectly-spherical cloud of flies as being controlled or directed but won't necessarily guess its source.
Intelligent opponents who have been exposed to standard magical effects, however, will likely suspect that not only is someone controlling the cloud, but that you are at its exact center - negating the advantage of Hiding to eliminate your sound. While they might not be familiar with this specific warlock invocation, as @Shadowmew says "Magical effects that obscure sight or other senses are not rare on the battlefield and can be a good indication that an enemy is nearby". And most such effects are centered on the caster - insect swarm, a portable darkness, spiritual guardians, a marine layer, etc. Other obscuring effects that are not centered on the caster (eg., fog cloud, cloud kill) tend to move with the wind. If there is a perfectly spherical obscured area that moves at a walking speed irrespective of any wind, it is a very good bet that there is a caster at the center who can be targeted with disadvantage.
@Eddymage says that having your opponents deliberately target the center of the cloud is "reasoning is based on false assumption and on metagaming". Certainly it would be if it was done by every opponent in every instance2. But if your intelligent opponents have experience with similar magical effects, and they can clearly see how your cloud looks and moves, then targeting you while you are unseen in the center is not metagaming, it is just good worldbuilding as it acknowledges that NPCs have history, too.
Is it better to Dodge?
Dodging is better than Hiding, yes! Because you will likely not be able to utilize the benefits of Hiding other than the one you already have by being invisible except perhaps moving silently.
Is Dodging better than being invisible? Somewhat. Dodging means your attackers are at disadvantage, but you already had that with invisible. However, if you Dodge you will in addition have a bonus to some saving throws.
1 As written, the aura "looks like buzzing flies". How do buzzing flies look - and in particular, how do they look different from non-buzzing flies? My inclination would be to rule that the flies actually are making noise as well. But suppose your DM follows a strict reading of the ability - now you are surrounded by a cloud of flies that look like they should be buzzing but are not making sound - hence making the whole thing even more suspicious.
2 Real metagaming takes it up a level, though: spell effects that target a point in space are centered on the intersections of grid lines, while those that target creatures are centered on the centers of squares. If your opponents are tracking that distinction than they are metagaming.