+1 for adding exhaustion. However, simply adding it to the game-as-is would be useless, because there are no rest limitations. Why would you ever care about exhaustion if you don't gain anything by reaching the point of exhaustion and/or "pushing through the exhaustion" to adventure more? Currently I suppose you save on food supplies, but given the vast quantity you're given that's irrelevant.
It would need to be combined with some rest limitation (in the form of restriction or discouragement). Obviously a simple restriction is "You cannot rest until at least one of your characters is exhausted," but even I'll admit that this is too punishing. You could very well end up in a state where you can't progress, say if you only have boss encounters left but are ~out of resources and just barely before the point of exhaustion.
Another solution is to heavily limit food supplies. This would make you want to rest less, but exhaustion would effectively be the point where the game tells you "it's okay to rest now. We promise that you'll have enough supplies if you rest at this frequency."
And my current favorite solution, is that certain companions should yell at you if you rest without them being exhausted. Lae'zel wants to rush to the creche. Lae'zel will think you're weak and/or not following her wishes if you rest when neither you nor she is exhausted. This discourages rest without limiting it.
These are all just examples: my main point is that exhaustion is a ~useless mechanic if there isn't some benefit for reaching the point of exhaustion. RPGs should be full of decisions and trade-offs, not strictly punishments for playing a certain way.