I don't think so, a rubber lip serves as a closure. The soap also needs the right consistency, disinfectant would probably drip through.

We use it with disinfectant and it works perfectly.

You can - it doesn't drip. Depending on the product, however, it dries up a bit, so that the next time I use it, the product squirts out the side. So I wipe the rubber lip lightly with my finger after each use.