I would take a buggy -if you've got a three-wheeler off-road type buggy I would actually take that. McLaren-type buggies are hopeless in the snow and airlines will take bigger buggies provided they fold flattish. Back carrier and suitcases....arghh. Would be heavy going. Plus, you can hang your ski boot bags off the buggy :)
I would put fashion to one side and take only one coat each - e.g., just your ski jackets if you can bear it. then you can wear them on the way out rather than have bulky things hogging space in your case.
Take lots of thin cotton layers for your baby - think leggings and long-sleeved tee shirts, an all-in-one snowsuit for him too, preferably wiht attached mittens and booties. I would also recommend ditching baby socks for the holiday and putting him in tights. Warmer and none of that pesky looking-for-the-missing-sock boredom. I am sure he's man enough to handle it :) He won't need that many clothes really, and if you run out then baby clothes are easy to wash in a sink and dry on radiator overnight.
Are you going with a specialist family operator or is the nanny a bolt-on ?
Nappies and baby formula etc seem to be £££ on the continent, if you can fit in your bags, may be best to take them with you. If you don't mind paying a bit more and your babe i sn't fussed about formula brands, then you can wait till you get there. Take some formula tho, france has funny laws on trading hours and you don't want to get there with a hungry infant to find all the shops are shut for some holiday/ because it's Sunday etc.
Use your ski boots/ helmets for storage en route - hand baggage limits are quite generous so make sure you fill 'em up. You might even be able to get a formula carton in a ski-boot if you try really hard 
I don't really think you have to worry too much about flying with your baby boy, I have never found it too traumatic to be honest - there's loads to look at for your baby. He'll be entertained and other passengers will likely want to interact with him unless you get a particularly sour crop on your plane. Dummies are useful for take off and landing but again, never had a problem with mine and neither have any of my friends/ family.
Have a fab time - bet it will be great.