The detective character you're describing sounds a lot like the character from the Super Solvers: Gizmos & Gadgets series. I've linked a video but not sure if it's the exact game you're thinking of
It's definitely it. I think the overhead light I'm remembering is from Mission T.H.I.N.K., but I do remember the Gizmos & Gadgets levels. Thanks for finally quashing this curiosity. I can die in peace now.
https://youtu.be/OzXHI22IOSA?t=113
edit: This DOS game might be the one you're thinking of
https://youtu.be/Wei0C21LBNM?t=28