Disappearing advanced search

On my page directestateagents.co.uk/property/ I have two instances of the search widget: a basic one without any filters, and an advanced one hidden behind a collapsible JS tab.
It’s what the client wants and it looks great, but unfortunately it doesn’t work when I enable the “For Sale / To Rent” filter buttons. When clicking either of them, the advanced search drop-down filters disappear, and there’s no way to bring them back unless you refresh the page.
I am guessing as far as JS is concerned, the two searches are two instances of the same object, hence the strange behaviour.
Is there a way to get the two widgets to co-exist happily?
I realize this is an unusual request, but I’m not great with JS so any help with this would be appreciated. Thanks