So maybe the next step, once tag is eliminated, is to re-do the letter frequency. It was, and now is:
cat 142 cat 131
dog 212 dog 211
has 141 has 131
max 241 max 231
dim 212 dim 212
Now we can eliminate cat and has since their frequency counts are the same.
That leaves dog, max, and dim. Dog has two unique letters (o and g); the other two only have one. So the person holding the letter g (or o) could turn his paper in. At that point, the person holding the letter d could not be sure whether the word was dog or dim. But once the second person stepped up, the third person holding the letter d could now be sure the secret word was dog.