You have discovered a major drawback of Hulu+Live TV. Some of the live streaming services won't run if you're away from your home network or location.
I don't have Hulu+Live TV, so I don't know for certain if this applies to that service, but many apps allow you to use Hulu as a provider. The TBS app (WatchTBS) may allow you to authenticate against Hulu as a provider. If so, you can use the WatchTBS app.
If not, then you may want to consider a different service. Many carry TBS, including Sling TV (half the price of Hulu+Live TV). You'd need to determine if the differences in services make it a cost effective alternative.
DBDukes Roku Community Streaming Expert Note: I am not a Roku employee.
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