Steampunk Adventure Idea

I'm writing an adventure for a convention. It's in Swedish, but I may switch to English. I dunno.

Actually there's a lot I don't know yet. This is what I have so far.

To the Moon, by Any Means Necessary!

Paris, June 14th 1873. The last 70 years have brought tremendous transformations and innovations. Steam-drive vehicles, airships, strange and wondrous artefacts. During a somewhat wild evening back in January some unnecessarily rich aristocrats decided that the next goal for humanity to strive for is the Moon!

To achieve this they're holding a competition with an almost unfathomable cash prize. Famous inventors of the world are participating, and are now gathered here in Paris. The start of the competition is tomorrow morning and the only rule is this:

My Problem

I've decided that the inventors participating, which is a mix of real and fictional characters, are these:

But. How do I involve the player characters here, and how should I play it out?

I'm thinking of starting off with all the inventors trying to recruit crews, and the player characters have to pick one to work for. Maybe in competition with some other team. Maybe old enemies in the mechanics guild?

There's a risk that characters want to run off in their own directions depending on specialities. Maybe I should play out the days where they get to handle fallout from external events and plan for the night, and then roll for how they do with their nightly activities of theft, sabotage, slander to journalists and whatnot?

I need a simple enough suitable mechanic for handling this, and some what to decide what the other teams are up to and are doing. Tips are very welcome.

-- CC0 Björn Wärmedal