In order to fix the issue, you obviously need to determine if user's input belongs to certain script(s). The most obvious way to do this is to use Unicode Regular Expressions:
var regexPattern = "[\\p{Katakana}\\p{Hiragana}\\p{Han}]+";
The only issue would be, that JavaScript does not support this kind of regular expressions out of the box. Anyway, you are lucky - there is a JS library called XRegExp and its Scripts add-on seems to exactly what you need. Now, the question is, whether you want to require at least three characters for non-Japanese or non-Chinese users, or do it otherwise - require at least three characters for certain scripts (Latin, Common, Cyrillic, Greek and Hebrew) while allowing any other to be searched on one character. I'd suggest the second solution:
if (XRegExp('[\\p{Latin}\\p{Common}\\p{Cyrillic}\\p{Greek}\\p{Hebrew}]+').test(input)) {
// test for string length and call AJAX if the string is long enough
} else {
// call AJAX search method
}
You might want to pre-compile the regular expression for better performance, but that's basically it.