If you don't mind a little modification to the crate.
You could attach a couple of 1" dimensional wood strips like forming stakes, under the crate, that just captures the sides of the bicycle rack.
Then using the same type of material, make a cross piece that will bridge across the bottom of the rack.
Drill two holes through each side where the pieces cross, and then use a carriage bolt and wing nut to clamp the crate to the rack.
The bolt holes would be on each side of the crate and not penetrating through any part of the rack.
View attachment 867238
Hopefully this crude drawing helps explain the idea.
Good luck!