How can I keep the case-sensitive box checked in Alkaline results?

You can use a regexp plus a $other option that inherits all other: values. You also have to look for opt:case in the searched text. The checkbox option input code will look like this (on the result template input form):

<input type="checkbox" name="OTHER" <!--SEARCH-GENERAL £other|HASopt:case~[checked] £post.search|HASopt:case~[checked]--> value="opt:case"> case sensitive</input>

This will simply produce checked when a case-sensitive search is performed.