CON save makes it more balanced, but DC15 seems a bit low.
As a 3rd level berserker barbarian, you would probably have con16 for a total save bonus of +5. This makes a 55% save chance, which still might seem a bit high to me.
As a 20th level berserker barbarian with the capstone, you would probably have con24, for a total save bonus of +13. This makes a 90% chance to not gain the exhaustion. I would say that you should aim to a 50-50 chance at higher levels with con20 (+11 bonus), so a DC of 21, and you would have a 25% chance as a level 3 beserker this way. I did this the same way as a save for exhaustion on my Barbarian Punisher Path, as a replacement for the beserker in a campain i wll DM in the future.