To sum up other contributions, the proposed construction:
is at least as secure as SHA-256 against collision attacks, that is the ability for an adversary to construct two files with the same hash; if SHA-256 was perfect, difficulty would be in the order of $2^{128}$ hashes.
is slightly less secure than SHA-256 against second-preimage attacks, that is the ability for an adversary to construct a file with the same hash as a given arbitrary file; if SHA-256 was perfect, difficulty would be in the order of $2^{256}/n$ hashes for $n$ separately hashed blocks (e.g. $2^{245}$ hashes for the envisioned $n=32768$), which is perfectly fine for any practical purpose.