Well, trying custom coding this is probably not something I recommend, so I would probably move to a single page if possible.
You can do preset the filter states though, this tutorial can help you with that.
Here you can also find some example for the asp_query_args filter. That one is to change the search arguments for the plugin. However in your case you want to change the arguments of the category archive page based on the query arguments of the search.
You could use the global $wp_query; variable within that filter, and then use the $args variable to replicate that within the $wp_query. That is not an easy task – as you have to detect which query is indeed the archive query, then pass all the arguments correctly.