Hi Jon,
A search results page is simply a filtered directory. You can create additional member directories, see:
https://dontdream.it/bp-profile-search/custom-directories/
and select one of them as the Target Directory in your search forms.
You can add one of your search forms to one of your directories using the option Add Form to Directory in the form settings, or using the widget Profile Search, see:
https://dontdream.it/bp-profile-search/
In short, to load results on the same page as the search form, you can create a new directory and add the form to that directory.