Will be very difficult to correctly export this figure correctly, too many different substructures that can all have their issues.
Make one Grid is much easier
array = Prepend[Transpose[Prepend[Correlation[Most /@ g[[1]]], h]],
Prepend[h, ""]];
Grid[Map[
If[NumberQ[#],
Item[Style[NumberForm[#, {3, 2}], 11],
Background -> ColorData["Temperature"][#]],
Item[Style[#, Bold, 11], Background -> White]
] &
, array, {2}],
Frame -> {All,
All, {{1, 1} -> False, {1, 2} -> True, {2, 1} -> True}},
ItemSize -> {4, 4}, Spacings -> {0, 0},
Alignment -> {Center, Center}]