What you're looking for is an Internet Bridging App
Easy on a PC
I think "Net Share" app does it on a 'Droid
Phone logs on to Gym then you approve T&C on phone browser
The app opens a separate "bridged" network to your DAP
Should work...
But with 512gb of in device storage, I can listen to recorded content for an hour a day
The other limitation is that Bluetooth doesn't have APT-X or other high density extensions.