Quetico permits are available online up to five months prior to entry date. Go
here, click the "Backcountry" tab, under which you select "Quetico" on the first line and select "Quetico" in the Park drop-down menu (why you have to select "Quetico" twice is way beyond my ability to understand, but if you don't, you don't get into the proper permit system. Then enter dates, etc, and you'll get your permit via mail. I just applied for my RABC permit, and had in in hand the next day! Have a fantastic trip!
TZ
WOOPS!! Tried the system using your dates and discovered the May 17 restriction you describe. Have you tried to reserve by phone?
Reservation numbers.