Perhaps we could do the reverse? Provide search suggestions if the user explicitly tells us to by beginning their input with a special character. '?' perhaps.
Alternatively, although not intuitive, if the user enters focus with the search bar through a specific key-stroke: CTRL + L for a regular URL entry, and re-route CTRL + K to our web-bar, but with suggestions.
On the other hand, if you could make this work, I'd love to see it.