Seeking Alternate Proof Regarding Closure Of Recursively Enumerable Languages
The question actually is easy, provided you take the right definition. RE languages can be defined using a Turing Machine in two equvalent ways. As an acceptor which accepts words in the language and halts (and may halt or loop otherwise), or as an enumerator, which will write the elements of the language (probably in some weird order) onto the output tape (assuming there is a separate working tape).

Using the latter concept several closure operations of the type you mention are trivialities. If I want to shrink, I start my enumerator, and when it outputs a word of the language apply shrink.