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:

The first team to set a (living) foot on the Moon wins.

My Problem

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

Alfred Nobel. Inventor of dynamite. Maybe he's a drunkard in this scenario? I haven't decided.

Phileas Fogg. The Englishman who travelled the world in 80 days, and is all around very inventive.

Ada Lovelace. The first programmer and a mathematical genious. In reality she would be dead by this time, but in this world she's still alive and around 70 years old.

Nikola Tesla. Young but already famous mad scientist. I've decided that this is a woman in this world, because it's such a sausage fest otherwise.

Impey Barbicane of the Baltimore Guns Club, who has nurtured an idea of firing a crew to the moon with a cannon for many years. Might gender-swap this one too. I haven't decided yet.

Professor Otto Lidenbrock, famous for travelling to the core of the Earth and coming back. He figures he's gone down already and now it's time to go up!

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 ew0k, 2021-10-11