Hi!
If the same exact search bar is placed to the widget, on the results page, then the options will be transfered as well. Different plugins/solutions will most likely not recognize the posted values.
Otherwise the only way is to custom code it, which I would rather not recommend, as it could get very difficult. Transfering the values from one plugin another is usually not an easy task, from my experience. I usually recommend a custom code snippet or something similar if possible, but in this case unfortunately a custom code would not help.