Installing a Bath Fan Without Attic Access?
Are you going to install a roof vent? If so and the ceiling is only 1' away, you can probably reach in through your 4" roof vent hole and screw into a ceiling joist or rafter tie to mount.

If you are going to vent directly into the attic and avoid cutting the roof open, you may run into problematic mold issues. If you're determined to go this route, your technique of screwing through the enclosure or drilling new holes should hold the fan adequately. The enclosure, aside from the electrical junction box, can take the screws. Remember you can and likely need to pop out the fan to get good access.