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My naive guess is that your requirement, for additivity (e.g. you want that the fingerprint of the sum (of parts) is derivable from sum of the fingerprints (of parts)) would conflict fundamentally with the requirements, of uniqueness and non-invertibility, of message digest/fingerprint algorithms, so I would look elsewhere for your kind of function.

Otherwise, I think the objection would go, that one could basically split a file repeatedly into sub-parts, down to the minimum block size, where the number of possible fingerprint values for each block is much smaller and hence the number of possible original different blocks is smaller too, so the original can be guessed (inverted) by brute force or something, and then be able to work back up from there, using your combinatorial/additivity property to reconstruct the fingerprint for larger and larger parts of the file, eventually compromising the secureness in message digest terms (e.g. uniqueness and non-invertibility) of the algorithm.