Thank you dop2000 for checking it out! :)
As you suggested I tried to change the stepping parameters but I didn't distinguish improvements, so for the time being I'm letting it as is.
Weirdly, I felt that reducing the number of rope parts, making them larger, and increasing their density actually made the rope break more easily... It probably needs more testing though.
However the pole suggestion was super useful and unlocked the situation in a major way!
I reworked a LOT of things, included the way I compute the rope's length, and now I can determine which is the closest pole in contact.
I've also stopped increasing the density of the bot, instead I change the force applied to the bot.
Since we now know which is the closest pole it's possible to apply a handicap (relative to rope's length) or not depending on the angle we move. If the rope is at max length we can still drive in reverse mode for example.
This is coming together, although the controls of the robot are still really bad... I wish I could simple add the "Car" Behavior to it but it doesn't go well with physics.
I'm still very interested in how I could make this prototype better (be it in its driving or rope mechanic) :D