Run throughs with the music are actually a problem. We are 14 who are competing, and the coach controls the music. If we get ours played more than once a session, it's a BIG DEAL. Plus we are 18 who are on the ice, most of us silver+, and a few prelim, so there's not much room, and actually doing those laps at full speed is complicated. . The sessions are too crowded but that's another story for another thread.
Did you join the French skating organization, or are you skating in an ISU competition for the U.S.? Or is it just a rink or coach-wise competition?
Hopefully, your competitors are in the same situation, and have the same issues, unless some of them own private ice.
So you need more ice time... In the U.S., most rinks have uncrowded mid-day sessions (except, maybe, around lunch time). Is it different in France? Or does your work schedule not allow that?
They have a bunch of famous outdoor rinks in France, right? Could you stand using one?
I've known skaters who used synthetic ice rinks to increase their strength and endurance. Yuk! - but they say it works. Wears out the blades, too.
If all else fails, some rinks rent private night-time ice. E.g., the rink I used to work at sometimes had "lock in" parties for the skating club all night long. Expensive - you'd probably need to share the ice, but you can pick who you share ice with, if you do the organizing.
If you can't get enough ice time, it's hard to believe that anything is going to be as good as skating itself to increase your skating endurance. I've taken off-ice skating-specific training classes, but they emphasized strength and flexibility more than endurance. I don't know, but assume the "Ballet for figure skaters" classes do the same thing.
Obviously, since I don't skate at your level, I can't guess what is specific to that type of skating.
But you could always try Ballet, or other performance dance, to augment running. Social dance helps increase general endurance too - people frequently dance most of the night, in a lively manner, and are frequently quite exhausted by the end. It can be fun too.
Or any other exercise that involves deep knee bends and jumps. Trampolines? The big trampolines are surprisingly gentle on your knees and other joints, because you slow down over a distance of a few feet. Sort of fun, too.
For that matter, no fancy equipment or cost required, a lot of competitive runners add high step intervals into their training regimens, but you need a fairly soft surface, if you don't want to mess up your joints.
The only exercise machines I know of dedicated to increasing skating endurance are designed for hockey, so maybe they aren't practical for you. But if pure endurance is what you want, some hockey training centers have them.
Surely there are coaches and other skaters in your area who you could ask what is available there?
Good luck! I hope you do well. Maybe you could share videos of your competition, and we could watch you win.