I’ve looked into this a lot after last year’s trip to Les Arcs.
I’m sure you’ve spent too long putting details into transfer websites who drag you in with tales of €50 transfers, let me save you the hassle – I tried them all, they’re all winter ones. There aren’t any scheduled transfers between Geneva and Les Arcs as far as I could find anyway – you can book one, but it’s expensive – €300 or more.
I personally couldn’t find any useful flights to any of the other 2 local airports in France in summer, loads in winter, none I could find last year in summer. But I was looking at Cardiff, Bristol and Birmingham which meant Geneva was my only option.
You can take a train or trains from Bourg to Geneva Airport, it’s not direct and not horribly expensive – €50ish I think from memory, you can use one of those clever websites to work out what tickets you need and then just make sure you get off and on when you should, but they don’t run either very early or very late so you might find you can’t make a fight or will be spending a long time in the airport – this was the only palatable option I found until the 11th hour.
In the end a mate decided to fly with me – which meant a hire car for a week was an option as our other mates took our bikes and bags in the van. In your shoes I’d be looking at a 1 day rental dropping the car off at Geneva airport on the French side, giving myself plenty of time in case I had to strip the bike to get it into the back of a hatchback and then cramming it in the bag at the airport. Geneva always has loads of flights you should be able to fly to Manchester fairly easily.