How to remove any words containing two adjacent characters with different in both cases and letters?

slist = StringJoin /@ Permutations[Characters@"ABCabc"]; 

pat =  a_ ~~ b_ /; ToUpperCase[a] != ToUpperCase[b] && UpperCaseQ[a] != UpperCaseQ[b];

Select[StringFreeQ[pat]]@slist

{"ABCcab", "ABCcba", "ABbacC", "ACBbac", "ACBbca", "ACcabB", 
"AabBCc", "AabcCB", "AacCBb", "AacbBC", "BACcab", "BACcba", "BAabcC", 
"BCAabc", "BCAacb", "BCcbaA", "BbaACc", "BbacCA", "BbcCAa", "BbcaAC", 
"CABbac", "CABbca", "CAacbB", "CBAabc", "CBAacb", "CBbcaA", "CcaABb", 
"CcabBA", "CcbBAa", "CcbaAB", "aABCcb", "aABbcC", "aACBbc", "aACcbB", 
"abBACc", "abcCAB", "abcCBA", "acCABb", "acbBAC", "acbBCA", "bBACca", 
"bBAacC", "bBCAac", "bBCcaA", "baABCc", "bacCAB", "bacCBA", "bcCBAa", 
"bcaABC", "bcaACB", "cCABba", "cCAabB", "cCBAab", "cCBbaA", "caACBb", 
"cabBAC", "cabBCA", "cbBCAa", "cbaABC", "cbaACB"}

Maybe this?:

DeleteCases[StringJoin /@ Permutations[Characters@"ABCabc"], 
 s_String /; 
  StringMatchQ[s,
    ___ ~~ a_ ~~ b_ ~~ ___ /;
     ! LowerCaseQ[a <> b] &&
     ! UpperCaseQ[a <> b] && 
     ToLowerCase[a] =!= ToLowerCase[b]]]