Using NAM 4x or 5x, I do not see a way to include the CID in the JWT.

1. Is it named something else in NAM?

2. Is there a way to get this value in there?

Here's a snip of the spec. 

*4.1.9. "cid" Client Identification Data Claim*
The "cid" (client identification data) claim allows the receiver of
the JWT to identify the entity that the JWT is intended to be used by.
The audience of the JWT MUST be able to identify the client with the
value of this claim.
The "cid" value is a case sensitive string containing a StringOrURI
value.This claim is OPTIONAL. If the entity processing the claim does
not identify the user of the JWT with the identifier in the "cid"
claim value, then the JWT MUST be rejected. The interpretation of the
registered to value is generally application specific.
A typical example of a registered to claim includes following: *
client_id that the audience can use to authenticate and
  identify the client.* A base64url encoded JWK. * A URL that points
to the key material that the audience can use to
  authenticate the user of the JWT.
*4.1.10 "cit" (Client Identification Data claim type)*
The "cit" (Client Identification Data claim type) identifies the type
of the "cid" claim. It is a StringOrURI value. 

Example:

{
  "ver": 1,
  "jti": "AT.0mP4JKAZX1iACIT4vbEDF7LpvDVjxypPMf0D7uX39RE",
  "iss": "https://{yourOktaDomain}/oauth2/{authorizationServerId}",
  "aud": "">https://api.example.com",
  "sub": "00ujmkLgagxeRrAg20g3",
  "iat": 1467145094,
  "exp": 1467148694,
  "cid": "nmdP1fcyvdVO11AL7ECm",
  "uid": "00ujmkLgagxeRrAg20g3",
  "scp": [
    "openid",
    "email",
    "flights",
    "custom"
  ],
  "auth_time": 1467142021,
  "custom_claim": "CustomValue"
}