Can I use a hygiene sling in the hospital bed to help clean my husband up?

Try to get an OT to come into your home to help you, they can see your set up and what is and isn't feasible. They should also train you on safe use of equipment and techniques to make your life easier.

What are you calling a "hygiene sling" If it is the sling for a Hoyer Lift if it is the one with a commode hole and it is the mesh one that is what they are intended to be used for as well as showering.
It might be easier to just roll him from one side to the other cleaning him and changing sheets at the same time. Doing this is actually faster than rolling him into the sling, hooking it up and then hoisting him. Particularly since the sling may become soiled as well and that may transfer back to him or to the bed.
If you have a Hoyer Lift there is a product called Lavin Lift Straps that go on each leg then they attach to the Hoyer Lift and it will lift the lower portion of the body so cleaning is easier but it does not make changing the sheets easier.