The dilemma of what to do about resting is a big, difficult to solve issue without a human DM. A human DM knows if you're long resting because you're almost dead, or if you're almost full up.
Just saying "you're too close to the enemy to long rest" is valid, but if you limit it to certain parts, it merely adds pointless extra walking to the game to leave the no-rest zone and after the rest to get back into the dungeon..
Random encounters sound at camp sound reasonable... but because of the nature of long rests, as the way to interact with the entire party and swap people back and forth, that means the camp will be crowded in Act 1. Count up the number of characters who could be in your camp. Right now there's a possibility of 5 Origins and 0-4 Custom Characters. That's 9 full-fledged PC's.
If a random encounter is not going to be a waste of time, it'll need to be at least somewhat threatening. How many enemies will need to be attacking to threaten 9 PC's? And this is not considering the possibility of the not-yet-added Origins, or other camp followers. Any combat would take forever to resolve.
The best I can think of is it consuming food or rations as a resource to limit long rests. Perhaps camp rations you can buy which scale up in cost with level?
