As written, no.
The Trickery Domain cleric's Invoke Duplicity Channel Divinity option (PHB p. 63) says:
... you can use your Channel Divinity
to create an illusory duplicate of yourself.
As an action, you create a perfect illusion of
yourself that lasts for 1 minute, or until you lose your
concentration (as if you were concentrating on a spell).
Let's apply this to your questions:
Does my cleric have any control on what the duplicate from Invoke Duplicity looks like?
It is a "duplicate" and a "perfect illusion of yourself" which means it looks (and sounds, tastes, smells etc.) exactly like you.
I know it's the same body, but can the clothes be changed?
If you were at my table this would extend to the clothes you are wearing and the things you are carrying, however, you may be able to convince your DM that you can make it appear in a different outfit etc.
Would my duplicate need to take a moment to change clothes or apply makeup — or is that even possible?
This is not possible. It is an illusion (specifically a phantasm) and has no real substance - think of it like a perfect hologram that you can smell, taste, hear and touch as well as see. It is objectively there; that is everyone can point to it and agrees what it looks like but it isn't real.
Illusions can interact with the real world in the ways and to the extent that the particular ways that the specific effect that creates them says they can. As illusions go, this one is pretty substantial, the rules describe it as:
- Perfect - for mine, this means that it is impossible to determine which of the two of you is real. This means that it will move and react realistically: if someone slices it with a sword it will bleed, if a wolf tries to smell it, it smells just like you and if an enemy wants to move through its space, they can't.
- lasts for 1 minute - it can't be "disbelieved" or seen through except by an effect that allows such (e.g. truesight)
As to what it can do: it can move as a bonus action, you can cast spells as though they originated from it (save that you must use your own senses) and if it and you are within 5 feet of a creature they have disadvantage on attack rolls against you (I would rule that they would also have to pick which one of you they want to attack).