I very much like your statement of the problem. As Maximuuus mentions, though, I feel like the solutions might create new issues and none of this addresses the resource management part of the problem. I also think that, though a story solution and a gameplay solution both need to happen, that they need to work together in one coherent system. I would either search for two separate solutions that can be woven together into one or look for a single solution that addresses both problems.
My ideas for the moment:
1) Eliminate the permanent camp entirely.
Putting down roots doesn't feel like something that you would do as a small group that has landed in the middle of nowhere and needs to go out searching for answers that might be very far away.
You need to do something with the companions that aren't actively in your party. They could go about their business in the world (where you could seek them out if you want to trade them in) or maybe they're following the main party, carrying all of the camping gear.
2) Camp wherever you are, but have risks commensurate with your location.
You can take a long rest or a short rest whenever you want. The location of that rest would be where you currently are or somewhere nearby. Someone could make a survival check, with higher results on that check corresponding to locations that are safer, easier to camouflage, or have better access to food/water/firewood. Having someone with a high passive survival in your group might point out good resting places as you wander around. In a populated area, like the grove or a city, you could initiate a rest by talking to an innkeeper or equivalent, or just take your chances on the street.
The risk needs to be high enough that you can't just rest all the time safely. Wherever you are, there would be some chance of a random encounter appropriate to the area (e.g. wandering monsters in the wilds or pickpockets in a city alley), but you can better prepare yourself for those encounters by setting a watch and rolling a perception check. Maybe if you're resting too close to the goblin camp, there's some chance that their scouts notice you and don't attack, but the camp is better prepared for you when you arrive. Lots of options here.
Odds of an encounter increase based on things like:
-how dangerous the area is
-how good of a campsite you found
-if you have food/water with you or if you have to go out looking for it (and how much is available nearby)
-how long your rest is - long rests in a dangerous area should be much riskier than short rests
-how long you have been in the area - the longer you've been around, the more likely your group has been noticed
-how good your watchperson's perception checks are (this could also impact whether you are surprised if you are found)
3) Have some representation of time passing.
This could be a day/night cycle where resting advances the clock a certain amount of time based on how long you rest. It could be the current system where it just turns to nighttime every time you take a long rest (though I personally don't prefer that). It could be other things in the world changing while you are resting. Again, lots of options and I don't know that there's a best one - just something that works with the rest of the system.
4) Decouple story moments from long rests.
If Wyll likes something that you did in battle, he can tell you about it right after the battle, wherever you happen to be. Or some amount of time later when there is no active threat nearby. Or maybe it's the next short or long rest. Maybe some are tied to a location - it would make sense for the party to take place in the grove, for example, no matter which side you took.
