This is exactly the solution I'm imagining. You do a take home after the initial soft screen(s) but prior to coming on-site. Onsite the focus is on the work you did for the take home--discussion like "why did you model X that way?", "if a new requirement Y was added, how would it impact the design", etc. And of course all the normal soft talk and cultural stuff. Just don't put them in front of a board and ask them to solve something under pressure, please.