Hi,
Thanks for your kind words again!
The way I would handle that is similar to what we have on our demo. For example, this is the “General” category archive for the post type results.
As you can see, there is a search bar on the top right corner, which will automatically filter the results. You have it right, for that you need the results page live loader feature.
I quickly looked at the archive source, and based on that I recommend using this DOM selector: .main-archive-loop
Instead of using the same search bar, I would create another one, without the filters. The reason is, that on the category archives when the archive live loading is enabled, the results are automatically restricted to the current category.
In my opinion the best practice is to use a search bar, without filters as the user already knows that they want to search only the given category archive.