That will not be a fun drive.
We did Edinburgh to Morzine in a diesel Primera. Took 22 hours, Nice is 5+ hours further on. We slept in turns and on the ferry, not sure we could have done much further without a proper stop.
On top of that you’ll have a lower speed limit in the van (60 on the motorway) and a lower cruising speed than we sat at.
Fuel will be around 30mpg if you’re careful in that big a van plus you have road tolls in France that a large van may be a bit more expensive than a car.
Rough guess you’ll be around £600 in fuel (1400 miles each way) £100+ in tolls, £140+ for the ferry (car was £40 each way I think).
Easy Jet charge £50 return for the bike on that flight, takes 2.5 hours from Edinburgh. Unless you’re carrying 15+ bikes I’d be very surprised if a van works out as the cheapest option before even factoring in your time.