BA is one of very few airlines that does not have a check-in the French side of Cointrin. As both France and Switzerland are in the Schengen area (despite Switzerland not being in the EU) there are no controls on movement between the French & Swiss zones, or for that matter at any other crossing point on the French/Swiss border - but that does not necessarily mean that it is easy to do so. To get from the Swiss zone to the French zone is easy as walking through a door but the reverse journey is considerably more difficult; you leave the French zone along a narrow corridor, then turn around and descend a narrow flight of stairs (a real pain with suitcases or push chairs) then walk through some more corridors to emerge at the back of the baggage hall (without the need to pass through Swiss customs), where you exit the Swiss arrivals hall, and to get to the Swiss check-in zone you have to go up a floor to the level you started from. If you are travelling with small kids and/or baggage it is not worth all the grief to save a few quid. Believe me; I have done it.

Another point is that any car you drive on a Swiss motorway (including the short stretch around Geneva to the airport) must display a toll 'vignette'. A Swiss-registered hire car will come with one whereas a French-registered hire car almost certainly will not. It is not worth risking taking an un-vignetted car on a Swiss motorway as the fine will be severe (remember, we are talking Switzerland here) and the cost of the vignette will eat into any savings on the hire car that you may make. You can avoid Switzerland altogether by taking the French Route Nationales from St Julien to Valleiry to Fort l'Ecluse where you double back to Fernay Voltaire and the sealed road under the runway to the French terminal, but that will add at least an hour to your journey time.

It's your call, but all-in-all I find that any marginal savings in car hire costs are just not worth the effort.