You are forgetting about the strapping between the ceiling and the joist. 7/8" below the joist is still above the edge of the ceiling. If for some odd reason this isn't the case raise the housing until it is flush or a little higher then the edge of the ceiling.
Also the hole you make for the housing should be completely covered by the grill when completed so there should be no need for sheetrock repair.
The bad news is you may run into a problem removing the old fan if you do not have access from above. Usually the fans are nailed to the joist before the ceiling goes up. You will not have easy access to the nails from below it. Your best bet may be to push up on the old housing so that the edge of the housing away from the joist is above the sheetrock and then try prying it off the joist. Just guessing though since I'm not there.
And make sure you vent it to the outside!