Don't get fixated on the 'travel' part. There's no reason why a standard wireless router couldn't have client mode (also sometimes called Media Bridge).
DD-WRT does support client mode so that'd expand the number of routers you could possibly use.
Client mode is available on an Asus RT-N66U I still use as an AP. It may be there because it's running the Merlin firmware but that's available for many models.
How badly does it work on 2.4GHz? You aren't trying to stream 4K so it should be possible. You may be better off trying to get 2.4GHz working well enough instead of struggling with 5GHz. You could still maybe go down the client mode route but do it on 2.4GHz and hopefully get a better signal than the Firestick manages on its own.