But the test fails despite Sort works (anti-konform to the character code)

In[1]:= "+" < "-"
Out[1]= "+" < "-"

In[2]:= {"+"} < {"-"}
Out[2]= {"+"} < {"-"}

In[8]:= ToCharacterCode["+"] < ToCharacterCode["-"]
Out[8]= {43} < {45}

In[12]:= weitzerOrder[ToCharacterCode["+"], ToCharacterCode["-"]]
Out[12]= True

To get these in ASCII order, you will have to write your own function.

In[18]:= Clear[WRule]
WRule = {"-" -> "WMinus", "+" -> "WPlus"};
StringReplace[
   Sort[StringReplace[{"(+3k-1,-4n/3-3m+k-7/3)", "(+3k+1,-n-m+k)", "(+3k-1,-n-m+k-1)"}, WRule], OrderedQ[{#1, #2}] &], 
   Reverse /@ WRule]

Out[20]= {"(+3k-1,-4n/3-3m+k-7/3)", "(+3k-1,-n-m+k-1)", "(+3k+1,-n-m+k)"}

or if the minus should appear later

In[21]:= Clear[WRule]
WRule = {"-" -> "WLast", "+" -> "WFirst"};
StringReplace[
 Sort[StringReplace[{"(+3k-1,-4n/3-3m+k-7/3)", "(+3k+1,-n-m+k)", "(+3k-1,-n-m+k-1)"}, WRule], OrderedQ[{#1, #2}] &], 
 Reverse /@ WRule]

Out[23]= {"(+3k+1,-n-m+k)", "(+3k-1,-4n/3-3m+k-7/3)", "(+3k-1,-n-m+k-1)"}