How do you reorder a sublist of strings, within a list?

One way:

d /. x : {__String} :>
  SortBy[x, StringReplace[#, {"A5" -> 1, "I" ~~ _ -> 2, "A" ~~ _ -> 3}] &];

% === newd

True

If you look at the output of StringReplace you will see remaining StringExpression heads, which is less than clean, but because they are all the same they do not affect the sort order.

A variation:

d /. x : {__String} :> 
   SortBy[x, Characters@# /. {{"A", "5"} -> 1, {"I", _} -> 2, {"A", _} -> 3} &];

% === newd

True

Map[Sort /* RotateLeft, d, {-2}]
% === newd

{{{"A5", "I4", "A2"}, {"A5", "I5", "A4"}}, {{"A5", "I4", "A2"}, {"A5",
    "I5", "A4"}}, {{"A5", "I4", "A2"}, {"A5", "I4", "A4"}}, {{"A5", 
   "I4", "A2"}}, {{"A5", "I4", "A2"}}}

True

 Map[ReverseSortBy[#, {StringCases["A5"], StringCases[RegularExpression["[I\d]"]]}]&,d,{2}]

{{{A5, I4, A2}, {A5, I5, A4}}, {{A5, I4, A2}, {A5, I5, A4}}, {{A5, I4, A2}, {A5, I4, A4}}, {{A5, I4, A2}}, {{A5, I4, A2}}}

%==newd

True