The website is awful. It's their major marketing tool and its badly constructed. It doesn't bode well at all.
As for the lounge, yes I agree that a dedicated lounge for one flight a day is probably a cost too far. It could be a lounge that's used by other carriers at other times of the day? If I'm paying the prices they're charging I expect something which is a little better than something which anyone can buy and use if they choose.
I wish La Compagnie well but I've seen nothing yet to suggest that they have their product offer right.