Just run a tap through the bolt holes on the bumper so the fairlead bolts thread into the bumper instead of putting a nut on the bolt. You may have to get bigger shorter bolts or use some washers to space them out.

 

Liquid nails is an industrial adhesive. They make some that will hold crazy weight. The hawse failed is not really load bearing so I would think a strong adhesive would do the trick. May be a bitch to get off later but a pry bar and hammer would do it.