Best way to secure loose flooring
Yes lots of folks overlay , I do in some cases not in others. You may get a construction adhesive to hold those in place. Some of the tiles have a peel and stick backing and that usually works well on vinyl but most say for dry locations, you said the wet look was built in so I might try a water resistant adhesive like liquid nails. It should not take very much you don’t want to make bulges or have it bleed up through the cracks before it drys. There are some silicone sealant / adhesives that should really stick well to the vinyl and once dry water won’t bother that type. I focus on moisture because in a bathroom that has been a majority of the issues I have found.