Random thoughts off the top of my head.
1. More than once people have mathed and shown that the Frenzy barbarian was balanced without exhaustion, so I wouldn't worry about it.
I could see where there might be some sort of need for a cost, since, when combined with Reckless Assault, it's giving an additional attack at advantage without any additional risk. That's two attacks with advantage every round at third level. Still, 1 level of Exhaustion is too harsh.
So, I have two ideas around that:
Limiting Frenzy to a number of times per Long Rest equal to their CON mod (minimum 1). This might be the most in keeping with the style of 5e.
The other would be to have Frenzy use either 1 HD or cause 1 level of exhaustion. This might be in most keeping with theme of the Path of Berserker while presenting another way to cost for abilities.
2. Exhaustion is primarily an exploration/travel tool, where a condition that will be fully cured by a long rest is not useful. This will completely take that tool away from you. But that doesn't feature in every campaign, so as long as you are sure you will never take your campaign in that direction you are fine. Yes, I put it in such strong terms because it's well suited for that and removing it for no reason will absolutely restrict your toolbox for running those types of adventures.
This is something good to consider. Thank you for bringing this up. Especially since there does need to be a robust exploration tool, and one of the concerns I had would be making getting rid of the running out of supplies issue too easy.
I had been thinking of using exhaustion to add a little more teeth to getting hit with a critical or dropping to zero HP without the severity of the DMG or substantially complicating bookkeeping. And to also do do without making the characters completely incompetent till their next Long Rest (or three). But going with the Frenzy route, losing 1HD or graining a level of Exhaustion might actually be a better to route to go. Yes, it is still burning through HD, especially for character most in need of them, which I am glad you bring up next, because I had not thought about that part.
3. This is a meaningless house rule unless you plan on giving out exhaustion for more thing. So I'm going to assume that you are planning on actually giving out more exhaustion, adn therefore use up HD in removing it.. How are you offsetting the loss of self-healing, or are you planning on shorter adventure days (with all that messes with the balance between long-rest-recovery and at-will classes)?
I was thinking of going the other way. Long rests would require a full downtime day (which would act as a standard RAW long rest, restoring class abilities, hit points and hit dice) and require a "safe" location or established camp, but also giving 1 HD and Lv + CON Mod HP (min 0) per night's rest in the rough. The intent is to make it possible to Long Rest in the field with the proper supplies, it's going to slow the party down if they are traveling and open them to more risk if they are camping a site of interest.
You bring up a good point about balancing with the martial classes, since their primary resources are hit dice and hit points. This would be a HD tax for tanking damage, which doesn't seem fair. Maybe an easy CON save (DC 5 or 1/2 damage taken, whichever is higher) to avoid the 0 HD tax, This would still allow the average 1st to 3rd level Barbarian or Fighter to safely take 12 to 14 points of damage in a single hit without concern of they go down.