On the other hand, using String allows for codepoints that aren't single
characters.

    ... which suggests a hybrid approach: A char[] array for the
one-to-one mappings, with a special value like '\u0000' meaning
"I don't know; check for exceptional cases."

    /* Untested, uncompiled, unscrutinized, un to the Nth: */

    static char[] translateTable = new char[65536];
    static { /* initialize it */ }

    static Map<Character,String> weirdCasesMap = ...;
    static { /* initialize it */ }

    String translate(String old) {
       StringBuilder buff = new StringBuilder();
       for (int n = old.length(), i = 0;  i < n;  ++i) {
           char oldc = old.charAt(i);
           char newc = translateTable[oldc];
           if (newc != 0) {
               buff.append(newc);
           }
           else {
               String news = weirdCasesMap.get(
                   Character.valueOf(oldc));
               if (news != null)  // allows for deletions
                   buff.append(news);
           }
       }
       return buff.toString();
    }

    If deletions (incoming characters that map to nothing in the
output) are common, consider using two special codes in translateTable:
one meaning "Check the map" and the other meaning "Ignore this."