I agree, the ability seems poorly written.
I would not make it 1/day. I would instead make it cost 1 "Drink"-equivalent to initiate, and then make the character wait until after the exhaustion period ENDS, before being able to initiate a new Rage (which would also cost 1 "drink"-equivalent). The alcohol spent to initiate the Rage would still count for duration (1 round worth). Multiple rages could not stack benefits, of course.
That however is fair. Consider, since the Drunken Rage supercedes Drink like a Demon's benefits ... some of those benefits include the preserved AC and Saving Throw levels. So, your AC and saves would worsen based on the number of drinks still in the character's system, and the resultant lowered Dexterity, Intelligence, and Wisdom; this might make your DM less wary of that ability.
Even making it just once per ENCOUNTER would be better, IMO, especially given the added loss of AC/Saves mentioned above.