In AES-256-CFB, does using a pseudo-unique key circumvent the weakenesses of reusing keys?
Goal: Encrypt multiple messages using the same key


That's what an IV is for! You can mitigate all the dangers of key reuse by using a unique initialization vector. The reason they exist is so that you can encrypt multiple messages without needing to go through the expensive process of changing keys (whether by generating a new random, truly independent key or by expanding one key into a number of keys with an algorithm like HKDF).


  Hypothesis: If we use, as the actual encryption key, a reused secret key concatenated with a publicly known yet varying (and never repeated) part, would that prevent exploits based on key reuse?


Your entire scheme is excessively complex and I haven't analyzed it. What I can say though is that you usually can concatenate a secret key with a public counter or other unique identifier and still treat the key as secure. A vulnerability that allows an attacker to violate some security principle of the cipher when multiple keys are used that are all secret but which differ in specific known ways is called a related key attack. Only ciphers that are badly broken, such as RC4, are vulnerable when used this way.

There is a better, standard way to expand one key into multiple though, and that is HKDF. It is a hash-based algorithm which expands a single key into any number of unique keys that can be treated as independent. This is useful because you don't have to sacrifice any part of the key by turning it into a public value. You can start with a 256-bit secret and derive any number of 256-bit secrets.