Not necessarily. The thing to understand here is that there are many different electronic balanced circuits and they don't all work the same. The only thing that can be guaranteed is a transformer and then that still depends on the wiring.
Whether a Eurorack module has a metal bushing jack or a plastic insulated one the one thing that you can be sure about is that the screen connection will be the pcb 0V because there is nothing else it can be and that will be at a different potential from the screen the other end which should be connected to chassis ground.
Beware of manufacturers who think that balanced lines are +4dBu signal level. A balanced line has no restriction on the signal level, the only limit is the maximum level that an input can take and that is usually over +20dBu, apart from some low end devices.
Always try the TS->TRS cable first as it is the cheapest solution and works in most cases.Or, would it just be preferable to, as suggest, just use a TS to TRS cable, that was previously referenced, from the case to the mixer, and use a TRS to TS cable, per below, for going from the mixer to the case?
The second cable won't work with all balanced sources. Some balanced outputs can accept one side being shorted to 0V, others don't.
It really is a minefield and you have to treat all combinations of equipment individually. You will get a better insight if you draw the output and input circuitry rather than just the cable and then you'll be able to see the conflicts. There is no simple answer apart from make everything balanced.