I would say that there are some hurdles to overcome here which aren't self-explanatory.
Most notably is that you apparently need music for the figure skating, but this music needs to use their system, which you don't have access to.
I guess I would wonder why you don't have access to the system when you are training, and why they wouldn't allow you to use that system at any time.
Secondly, even with any system that can be suggested, it will still require someone to flip the audio to that source, which means ... they need to have access to the booth and know what they are doing to get you audio.
Third - I guess if this is important, then I would speak with owners/senior management about the problem you are having. You could buy all the equipment necessary to connect with a wireless (wi-fi) solution, but if they aren't willing to use that system when you are there, then it is wasted money.
There is always the option of using in-ear headphones if they don't have a solution for you that allows you to use their system.