Clarification regarding filtering the products

I have created a store for my client using minimog theme. There is a filter on the shop page. The top most filter is ‘availability’. By default all the products are showing irrespective of available or out of stock. But client wants to show only available products initially and user should see “out of stock” products as well when check the ‘out of stock’ filter. My question: is this feasible? Here goes the site link: https://alpinechefstore.com/collections/all.

Please help me if this is feasible.

Thank you guys