Is "Bit Independence" of SHA-2 hashes secure?
There is no known exploitable relationships between the words (32/64 bit "chunks") of SHA-2 but this is always a source of cryptanalysis.  There are some academic weaknesses against reduced round variants of SHA-2 but nothing against the full cipher.  Still hashes are cheap so why not just use multiple hash functions?

Seed = HASH(secret)  &lt;- This should be a KDF instead see below
A = HASH(Seed)
B = HASH(A)
C = HASH(B)


If you need less bits you can just chop them down (it is good enough for NIST). Here is another option.

&gt; Seed = HASH(secret)  &lt;- This should be a KDF instead see below 
&gt; A = HASH(Seed XOR ConstantA)
&gt; B = HASH(Seed XOR ConstantB)
&gt; C = HASH(Seed XOR ConstantC)


Keep the constants the same same length as the output of HASH to avoid a biased result.

Use a KDF

I am assuming "secretKeyNonce256bit" is really a human generated password.  Using a simple unsalted hash to derive a key from a low entropy secret is a bad practice. You say "not the case here" but it is unclear if you mean it is not a password or not a "weak password". If "secretKeyNonce256bit" is a 64 byte random value from a CSPRNG then ignore this but if it is the result of any human selection you really should be using a true KDF (like PBKDF2, Bcrypt or Scrypt).