The stronger way would be to cut a piece of drywall out that spanned the two studs to the studs on each side. About 12" high. The hole would then be 48wide by 12 high.

Get a piece of 2"x10" lumber that is 16" long. Place that horizontally between the studs where you want to hang your bike and nail/screw it into place through the studs on each side where you cut away the drywall so you have room to nail/screw.

Patch the drywall (5 minutes quickset will be your friend), paint. Project will take a 2-4 hours. Pro tip: take a small (4"x4") piece of the removed drywall to the local big box home supply to match the paint color. They do it with a machine that matches about perfectly.

That 2x10 will be the backer for your mount.