For the Paris to Courmayeur route, it is better to go via Geneva rather than Saint-Gervais; the journey is more relaxed with just one change, and the bus stop is only a 5-minute walk from the station.
Please note that your arrival time will depend largely on traffic conditions at the Mont Blanc Tunnel. For the connection to La Thuile, use bus route 573.
Make sure you check the timetables well in advance of your departure, as they are not displayed on site and local information is difficult to obtain.
Once in La Thuile, please note that the bus stops about 2 km before the campsite.
Finally, although it is possible to reach the refuge on the same day, it is strongly recommended that you set aside a full day for travel so that you can start the hike the following morning, well-rested and stress-free.