This should be the default behaviour. Choosing a Low Stock Activity of "Disable Buy Button" should disable the buy button when productVariant.StockQuantity < productVariant.MinStockQuantity.
The thing is I can't see that this check is made anywhere.
In this case I would normally add a property e.g. InStock to the underlying class and return its value as
bool Instock = (productVariant.StockQuantity < productVariant.MinStockQuantity)
Then on your product variant templates in code behind change line 127 to:
if ((!productVariant.DisableBuyButton) & (productVariant.InStock))
You can test easily without the property just by changing to:
if ((!productVariant.DisableBuyButton) & (productVariant.StockQuantity > productVariant.MinStockQuantity))
This will then disable the buy button if not in stock.
As you say, you may wish to change the behaviour so that it displays a useful message like "This item is not currently in stock" or redirects to a related products page.