I'd absolutely allow a barbarian to rage outside of combat.
There's no combat limitation to enter a rage. It's be totally fair to enter a rage to jump farther, climb faster, or smash down a door.
The catch is duration, as it would only last a turn. RAW unless they take damage or attacked a hostile creature, the rage ends. So even in a time of exploratory stress where the barbarian is trying to smash down the door of a flooding room before they die, they can't maintain a rage.
Now, this is where the advantages of 5e comes into play. Because the game is rulings and not rules, I can say that provided there is a source of stress and adrenaline, and the barbarian continues to be active each round, they can maintain a rage. If they're climbing frantically, than Athletics checks count, and if they're battering down a door those attack rolls count, and if they're competing in a tense arm wrestling match than as long as that goes on they can rage.
After all, the limitation is there so you drop out of rage if stunned, held, or generally inactive. It encourages you to always being doing something and not being overly cautious or readying your attack. Because that would be weird. It's not there to put a damper on creative uses if the DM permits.
If the barbarian player wants to spend a limited resource increasing their odds of success, then it should work. Because that seems like fun.
But I also have the freedom of the DM to shoot it down if it seems like the player is trying to take advantage of the rules. "You try and rage, but the tension isn't there and you don't feel the same sense of pressure or danger. You think you could rage, but it would be short lived."