Hi Kenneth,
Interesting.
I'm not 100% sure of the full solution. But if I was to try it I would put a SEBLOD form module on the search page (hidden) and use javascript to take the values that the user entered in the search form, and fill the fields in the form module.
That part is the easy part. THEN you'd need to find a way to link the buttons together so that when you clicked "search" it also posted the contents of the form at the same time... that I'm not sure about.
Good luck - would love to hear your solution when you find one.