Hi shinamae
Unfortunately cats are just something you have to live with in suburban environments really, you could ask your neighbour to keep them in but as you say that request may not go down too well and there is no guarantee other cats won’t come along at some point.
I don’t know the design of your hedgehog house but the trick really is getting the entrance hole just the right size and if possible using a piece of pipe at the entrance to make a tunnel that is either too narrow or too long for a cat to fit down.
Here is one I made from a plant pot with the bottom cut out…..
