Hostile Divisor Numbers
05AB1E, 12 10 bytes
µNNÑ€ÙSDÙQ

-2 bytes thanks to @Emigna.
1-indexed
Try it online or verify most test cases (last two test cases are omitted, since they time out).
Explanation:

µ           # Loop while the counter_variable is not equal to the (implicit) input yet:
 N          #  Push the 0-based index of the loop to the stack
  NÑ        #  Get the divisors of the 0-based index as well
            #   i.e. N=9566 → [1,2,4783,9566]
            #   i.e. N=9567 → [1,3,9,1063,3189,9567]
    €Ù      #  Uniquify the digits of each divisor
            #   → [&quot;1&quot;,&quot;2&quot;,&quot;4783&quot;,&quot;956&quot;]
            #   → [&quot;1&quot;,&quot;3&quot;,&quot;9&quot;,&quot;1063&quot;,&quot;3189&quot;,&quot;9567&quot;]
      S     #  Convert it to a flattened list of digits
            #   → [&quot;1&quot;,&quot;2&quot;,&quot;4&quot;,&quot;7&quot;,&quot;8&quot;,&quot;3&quot;,&quot;9&quot;,&quot;5&quot;,&quot;6&quot;]
            #   → [&quot;1&quot;,&quot;3&quot;,&quot;9&quot;,&quot;1&quot;,&quot;0&quot;,&quot;6&quot;,&quot;3&quot;,&quot;3&quot;,&quot;1&quot;,&quot;8&quot;,&quot;9&quot;,&quot;9&quot;,&quot;5&quot;,&quot;6&quot;,&quot;7&quot;]
       D    #  Duplicate this list
        Ù   #  Unique the digits
            #   → [&quot;1&quot;,&quot;2&quot;,&quot;4&quot;,&quot;7&quot;,&quot;8&quot;,&quot;3&quot;,&quot;9&quot;,&quot;5&quot;,&quot;6&quot;]
            #   → [&quot;1&quot;,&quot;3&quot;,&quot;9&quot;,&quot;0&quot;,&quot;6&quot;,&quot;8&quot;,&quot;5&quot;,&quot;7&quot;]
         Q  #  And check if it is still equal to the duplicated list
            #   → 1 (truthy)
            #   → 0 (falsey)
            #  And if it's truthy: implicitly increase the counter_variable by 1
            # (After the loop: implicitly output the top of the stack,
            #  which is the pushed index)