Three Indistinguishable Dice
Jelly, 22 20 bytes

6ṗ3Ṣ€ṢðQ€L€Ụịḷi’:6d6‘S


Try it online! or simulate all 216 outcomes.

Background

We map each unordered triplet of dice rolls (listed with respective multiplicities) to an ordered pair of dice rolls in the following fashion:

[1,2,3],[1,2,3],[1,2,3],[1,2,3],[1,2,3],[1,2,3] -&gt; [1,1]
[1,2,4],[1,2,4],[1,2,4],[1,2,4],[1,2,4],[1,2,4] -&gt; [1,2]
[1,2,5],[1,2,5],[1,2,5],[1,2,5],[1,2,5],[1,2,5] -&gt; [1,3]
[1,2,6],[1,2,6],[1,2,6],[1,2,6],[1,2,6],[1,2,6] -&gt; [1,4]
[1,3,4],[1,3,4],[1,3,4],[1,3,4],[1,3,4],[1,3,4] -&gt; [1,5]
[1,3,5],[1,3,5],[1,3,5],[1,3,5],[1,3,5],[1,3,5] -&gt; [1,6]
[1,3,6],[1,3,6],[1,3,6],[1,3,6],[1,3,6],[1,3,6] -&gt; [2,1]
[1,4,5],[1,4,5],[1,4,5],[1,4,5],[1,4,5],[1,4,5] -&gt; [2,2]
[1,4,6],[1,4,6],[1,4,6],[1,4,6],[1,4,6],[1,4,6] -&gt; [2,3]
[1,5,6],[1,5,6],[1,5,6],[1,5,6],[1,5,6],[1,5,6] -&gt; [2,4]
[2,3,4],[2,3,4],[2,3,4],[2,3,4],[2,3,4],[2,3,4] -&gt; [2,5]
[2,3,5],[2,3,5],[2,3,5],[2,3,5],[2,3,5],[2,3,5] -&gt; [2,6]
[2,3,6],[2,3,6],[2,3,6],[2,3,6],[2,3,6],[2,3,6] -&gt; [3,1]
[2,4,5],[2,4,5],[2,4,5],[2,4,5],[2,4,5],[2,4,5] -&gt; [3,2]
[2,4,6],[2,4,6],[2,4,6],[2,4,6],[2,4,6],[2,4,6] -&gt; [3,3]
[2,5,6],[2,5,6],[2,5,6],[2,5,6],[2,5,6],[2,5,6] -&gt; [3,4]
[3,4,5],[3,4,5],[3,4,5],[3,4,5],[3,4,5],[3,4,5] -&gt; [3,5]
[3,4,6],[3,4,6],[3,4,6],[3,4,6],[3,4,6],[3,4,6] -&gt; [3,6]
[3,5,6],[3,5,6],[3,5,6],[3,5,6],[3,5,6],[3,5,6] -&gt; [4,1]
[4,5,6],[4,5,6],[4,5,6],[4,5,6],[4,5,6],[4,5,6] -&gt; [4,2]
[1,2,2],[1,2,2],[1,2,2],[1,3,3],[1,3,3],[1,3,3] -&gt; [4,3]
[1,4,4],[1,4,4],[1,4,4],[1,5,5],[1,5,5],[1,5,5] -&gt; [4,4]
[1,6,6],[1,6,6],[1,6,6],[2,3,3],[2,3,3],[2,3,3] -&gt; [4,5]
[2,4,4],[2,4,4],[2,4,4],[2,5,5],[2,5,5],[2,5,5] -&gt; [4,6]
[2,6,6],[2,6,6],[2,6,6],[3,4,4],[3,4,4],[3,4,4] -&gt; [5,1]
[3,5,5],[3,5,5],[3,5,5],[3,6,6],[3,6,6],[3,6,6] -&gt; [5,2]
[4,5,5],[4,5,5],[4,5,5],[4,6,6],[4,6,6],[4,6,6] -&gt; [5,3]
[5,6,6],[5,6,6],[5,6,6],[1,1,2],[1,1,2],[1,1,2] -&gt; [5,4]
[1,1,3],[1,1,3],[1,1,3],[1,1,4],[1,1,4],[1,1,4] -&gt; [5,5]
[1,1,5],[1,1,5],[1,1,5],[1,1,6],[1,1,6],[1,1,6] -&gt; [5,6]
[2,2,3],[2,2,3],[2,2,3],[2,2,4],[2,2,4],[2,2,4] -&gt; [6,1]
[2,2,5],[2,2,5],[2,2,5],[2,2,6],[2,2,6],[2,2,6] -&gt; [6,2]
[3,3,4],[3,3,4],[3,3,4],[3,3,5],[3,3,5],[3,3,5] -&gt; [6,3]
[3,3,6],[3,3,6],[3,3,6],[4,4,5],[4,4,5],[4,4,5] -&gt; [6,4]
[4,4,6],[4,4,6],[4,4,6],[5,5,6],[5,5,6],[5,5,6] -&gt; [6,5]
[1,1,1],[2,2,2],[3,3,3],[4,4,4],[5,5,5],[6,6,6] -&gt; [6,6]


This makes all outcomes equiprobable.

How it works

6ṗ3Ṣ€ṢðĠ€Ụịḷi’:6d6‘S  Main link. Argument: D (three dice rolls, sorted)

6ṗ3                     Generate lists of length 3 over [1, 2, 3, 4, 5, 6].
   Ṣ€                   Sort each triplet.
     Ṣ                  Sort the list of triplets.
      ð                 Begin a new, dyadic chain.
                        Arguments: A (list of triplets), D
       Ġ€               Group each; group the indices of each triplet by the
                        the corresponding values.
                        For a triplet [a, b, c], this yields:
                          [[1], [2], [3]] if a &lt; b &lt; c
                          [[1], [2, 3]]   if a &lt; b = c
                          [[1, 2], [3]]   if a = b &lt; c
                          [[1, 2, 3]]     if a = b = c
           Ụ            Grade up; sort the indices of A by those 2D lists.
            ịḷ          Retrieve the elements of A at those indices.
                        This sorts A as in the previous section.
              i         Find the (1-based) index of D.
               ’        Decrement to get the 0-based index.
                :6      Divide the index by 6, rounding down.
                  d6    Divmod; return quotient and remainder of division by 6.
                    ‘   Increment to map [0, ..., 5] to [1, ..., 6].
                     S  Sum the results.