From Montreux to Genève there are two connections each hour with supplement-free IR train.
From Genève take as well a supplement-free TER to Lyon. Change there to a direct TGV either to Marseille or Nice. Other possibility would be a direct TGV Genève - Marseille/Nice but there are only two respectively one direct connections per day.
The TGV require a 3/10€ supplement (off-peak/peak) that you have to buy directly at a train station. I'm sure that you can do so at Gare de Montreux. Like you've said there's only a limited number of seats available for InterRailers on TGV so the sooner you do the reservation the better.
Possible connection on July 20th:
Montreux 1219 - 1324 Genève
Genève 1344 - 1717 Marseille/1955 Nice
Note that the direct TGV from Genève might be full quickly.
For searching for other connections via Lyon use the travel planner:https://en.rail.cc/interrail-train-connections