How do I add options to a built in function, which only apply to a certain class of argument?

Question

I'd like to add an option to a built-in function that only applies when it is given an argument of a certain form.

For example, something like this approach to extending DateString to support an extra option when it is given a date argument that matches the form zoned[...] (forget that zoned does nothing here):

DateString[zoned[x_], spec_, opts : OptionsPattern[{TimeZone -> 0,ShowZone->True}]] ^:= 
DateString[x, If[OptionValue[ShowZone],
   Append[spec, If[OptionValue[TimeZone] != 0, "UTC" <> ToString[OptionValue[TimeZone]], "Z"]], spec]]

Since ShowZone is not an option for DateString, I get an error.

"DateString::optx: Unknown option ShowZone in DateString..."

But I can't just associate TimeZone as an up value for zoned with

zoned/:Options[DateString]={ShowZone->True}

because that also results in an error:

Options::tag: Rule for Options of Options[DateString] can only be attached to DateString.

It seems I have to (unprotect DateString and) then add a new option globally, which is generally just ignored:

Unprotect[DateString]; 
Options[DateString] = Union[Options[DateString], {ShowZone -> False}]; 
Protect[DateString];

Is this the only approach? Even this approach leaves ShowZone blue in the function definition and red in function invocations.