I don’t think it would be unkind to say that this API is pretty bad at searches atm (it’s in Beta though right?). You cannot search based on any logical parameters (SIC class, locale, registration date) from what I understand. Your best bet is to download the company house Data from Companies House (yes all of it!) and wait until the new data is release (once a month) and compare the files using version control or some SQL trickery.
Unless someone else can suggest something better (I live in hope!) then this your best option.
Luke.