Unfortunately that is not possible. The plugin only works with it's own form, as it has a special structure and contains all the required information for the ajax request, lots of attached handler script functions and stylesheet rules as well.
The theme search has to be replaced, if you want to use the live search feature. Usually if it's possible, I implement something in future releases to overcome a certain issue, but in this case there is nothing I can do. There are too many themes on the market, all with different search boxes with all kinds of scripts and node structures, it is just impossible to make a solution that would work with even just some of them – and most of the search features would have to be disabled as well.
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