Can an adversary distinguish a private key from a pseudo-random string of the same length?
Usually choosing a safe password and standard parameter for the PBKDF2 key derivation would be enough protect your cipher. If PBKDF2 is correctly used, the symmetric key you get as output is well generated and attacking the ciphertext is infeasible.

Protecting a private key as you're doing is a standard operation, usually the password is used (in PBKDF2) to generate a symmetric key which is used with a symetric algorithm (let's say AES) to protect your precious private key. 
AES is supposed to be secure even if an attacker is given access to a decryption oracle (i.e.: it can get decryption of every ciphertext he wish but yours, the challenge). So modifiing the plaintext (stripping the header) would not improve the security level.

After your edit.
Let's imagine that someone saw almost all your password while you were typing it or know that it is included in a relatively small set.

You said that you would strip the "-----BEGIN RSA PRIVATE KEY-----" string, that means that the key is in PKCS#1 standard, so what you keep is a base64 encoded string corresponding the following DER structure:


RSAPrivateKey ::= SEQUENCE {
  version           Version,
  modulus           INTEGER,  -- n
  publicExponent    INTEGER,  -- e
  privateExponent   INTEGER,  -- d
  prime1            INTEGER,  -- p
  prime2            INTEGER,  -- q
  exponent1         INTEGER,  -- d mod (p1)
  exponent2         INTEGER,  -- d mod (q-1)
  coefficient       INTEGER,  -- (inverse of q) mod p
  otherPrimeInfos   OtherPrimeInfos OPTIONAL
}



(source)

So, an attacker could try to check for the value of $e$ (usually a standard value among 2 or 3) or check if $p \times q = n$, or ... 
So even if, in theory an RSA key is just a couple of integers, it is usually stored in a well known structure helping the attacker to recognize what he has found.