Interesting problem. I'd be a little concerned that an interview setting is not necessarily a realistic microcosm of a real project. You're going to enter with much less understanding of the problem and 'customer' than you might expect to have in a normal professional engagement. However since you seem to have some idea what a successful outcome would look like this should be manageable.
Everyone has their own idea what a good kickoff looks like so if you can frame yours in a context appropriate for the company you want to join that's probably better. All I can offer is my experience as a contractor at Carbon Five.
Your question refers to a couple of different activities; requirements gathering, project kickoff, story planning. I'll often spend 45 minutes on each of those so I would want to prioritize which of these you actually need to cover.
Normally I would start with a kickoff to explain the process my team uses, how it works, and what is expected/required from everyone participating especially the customer/project owner. That's probably not going to give you a good chance to highlight your abilities in an interview though so you may need to hope that can be taken as assumed knowledge or under a vague description: "this team will be using an agile iterative process to build your product, we're not going to plan everything today but we should leave in a good position to plan each feature as they come up."
I would focus on requirements gathering activities. Since you'll have a customer and developer present you can aim to produce some actionable stories but I would make it clear that in an initial meeting many of those are likely to be epics and need to be broken down in future conversations unless the problem is very simple. You should be able to get everyone to participate in a planning activity and give yourself a good opportunity to demonstrate your ability to lead an interactive discussion and discovery of the product the client needs.
I would try to start by sketching some very lightweight personas as the customer explains the problem. Just enough to have a reference for who you are building this for and what constraints you understand them to have. Normally this is something I would try to have beforehand but I think it's worth 5 minutes to show a shared understanding of who this product is for and if there are multiple user personas to agree to focus on one. (I've found the style Lane Halley describes in Personas for Coders useful.)
Given some idea who your user is I would try to build a simple experience map. In 15 minutes you should be able to sketch out the activities your user performs which interact with the product and the different touch points involved. This should be a good chance for the developer to help identify external systems you'll need to integrate with and for the customer to reveal interactions outside the bounds of the product you control (e.g. forwarding emails to other people).
Finally I would use the activities identified in that experience map to start writing some high level stories and defining some initial milestones for the project. I find that a story map is a good way to start organizing these sort of stories and shuffle them into functional (rather than temporal) milestones. This is also a chance for you to demonstrate writing good INVEST stories with a customer and developer.
I would certainly be out of time at this point but hopefully you can block off a few minutes to discuss next steps like detailed estimation of the first sprint or milestone and to ask about the process the interviewers normally use.