I devised a new distance-function df1 which is zero when the energies are equal, 1 when uncertainety-intervals overlap and the absolute value of the energy-difference otherwise.

df1[xx_, yy_] := Module[{},
  x = xx; y = yy;
  j1 = Interval[{x[[1]] - x[[4]], x[[1]] + x[[4]]}];
  j2 = Interval[{y[[1]] - y[[4]], y[[1]] + y[[4]]}];
  ds = IntervalIntersection[j1, j2];
  Which[
   x[[1]] == y[[1]], 0,
   ds === Interval[], Abs[x[[1]] - y[[1]]],
   True, 1]
  ]

This distancefunction applied to the 2-subsets of level52 shows that two energies are "similar", i.e. having distance = 1, and the others not:

In[17]:= df1 @@@ (Subsets[level52, {2}])

Out[17]= {1, 55, 101, 154, 229, 33, 79, 132, 207, 46, 99, 174, 53, 128, 75}

So level52 should be clustered, what surprisingly is not the case:

 In[15]:= FindClusters[level52, DistanceFunction -> (df1[#1, #2] &)]
% // Length

Out[15]= {{{6430, 0.93, 0.0808, 12}, {6452, 0.56, 0.112, 13}, {6485, 
   2.03, 0.0848, 15}, {6531, 0.78, 0.0579, 18}, {6584, 0.56, 0.0488, 
   21}, {6659, 0.83, 0.0483, 25}}}

Out[16]= 1

So I decided to do it on my own, and - as expected - find in level52 two clusters. One of two "similar energy-levels, and the rest of all others

In[29]:= len = Length[level52];
pairs = Flatten[Table[{level52[[i]], level52[[j]]}, {i, len - 1}, {j, i + 1, len}],1];
Select[pairs, df1[#[[1]], #[[2]]] == 1 &]

Out[31]= {{{6430, 0.93, 0.0808, 12}, {6452, 0.56, 0.112, 13}}}

Sorry, but it is by no means clear what you actually want to achieve.

What should your

Drop[level52, None, {2, 3, 4}] -> level52

do? In my system this just makes a Rule. Look at

FullForm [ Drop[level52, None, {2, 3, 4}] -> level52 ]

And then? What should be done with this rule?

Or do you want to eliminate some entries of level52? And if yes, which?

Is level52 one of your 200 or so energy-levels. Or do you want to cluster within level52?

It might be helpful if you gave more details, or a written example (in a notebook) what your intentions are. E.g. ten or 15 energy levels in a list, and the result you want to have.

Well, I would expect something like that:

FindClusters[{ {6430, 0.93, 0.0808, 12}, {6452, 0.56, 0.1120, 13}, {6485, 2.03, 0.0848, 15}},\\ first cluster, energies compatible within error 
   {{6531, 0.78, 0.0579, 18}, {6584, 0.56, 0.0488, 21}},\\ second cluster, energies compatible within error {{6659, 0.83, 0.0483, 25}} \\third cluster: energy not compatible within errors with others energies 
   }