Is there nuance in the ruling of recreational travel to non-Muslim countries? I'm planning a hiking and nature trip to Georgia, avoiding the cities, because it's visa-free for my travel companion, whose nationality makes getting visas elsewhere difficult. Is this permissible?

The scholars forbade travel to a non Muslim country in order to protect your religion from being influenced by them and their way of living, and that is the scale you measure your own case by. Imam Ibn Uthaymin made it conditional on having knowledge that repels doubts, religion that restrains you, and a need for the journey.