He has a rebel

I appreciate the stylebook's patience, but don't you wish -- just once or twice -- it would be a little blunter* with the questions that arrive at "Ask the Stylebook"?

Dear Bethesda:

No.

Love,
The stylebook

Contractions aren't a "grammar" issue. They're primarily a register issue, and in some cases you can fashion a pretty spectacular ambiguity issue from them, but you can't roll them up and hit your colleagues upside the head with them in the name of "grammar."

Contracted forms of "have" lie on a scale of cluefulness from "fine" to "gah." At the one end, there's no question that I've got a secret and I have got a secret are the same clause. The second is more formal, but formality isn't a function of grammar. Toward the middle, we sometimes vary by region: American English is pretty casual about shortening auxiliary "has," as in He's got an idea, but less so than our transmarine friends in shortening the main verb: I've an idea. 

Out at the far end is grammar: not a lack of it, but too much of it about too many things. Heather is a camera could become Heather's a camera with no problem, but Heather has two Uzis is not the same thing as Heather's two Uzis. (One's a clause, one is a noun phrase.) So much as I might prefer the shorter answer, the longer one is something like:

Dear Bethesda:

's is a perfectly grammatical shortening of has. In some cases, it may annoy your supervisor, so you should memorize those cases and be prepared to recite some swill about the evils of contractions that has nothing to do with "grammar." In others, the contraction might be grammatical, but since it's "grammatical" about some meaning other than the one you want, you should avoid it.

If you can't already tell the difference at better than chance levels, you should consider changing majors very soon.

Love,
The stylebook
 
* Bethesda, Bethesda, you have no complaint
You are what you are and you ain't what you ain't
... sorry