Is it permissible to deal with conventional, non-Islamic French banks to obtain the necessary financing, due to the unavailability of a Sharia-compliant alternative, and does necessity permit this or not?

The Islamic Sharia (definition) has forbidden Riba (usury/interest), considering it one of the gravest major sins. This prohibition is waived only in cases of extreme necessity where there is a fear of perishing or nearing perishment. Increasing income, developing trade, and investing are not considered necessities that permit dealing with Riba.