Prove: possible to construct automata accepting all strings of other automata sans 1-length strings
Basically, the answer is what @Shaull says in the comments: construct a product automaton with the one that accepts all strings of length unequal 1.
What we use here is the fact that regular languages are closed under intersection. The product construction takes two automata, and simulates them in parallel. States are pairs of states of the original two automata. The combination accepts if both components accept.

But you ask for "minimal operations". So, have a look at what is in fact the automaton that accepts strings of length not 1. It simply has three states "zero", "one" and "many" with a loop for each letter at the last state. It simply counts letters up to length two (and then keeps in that state).

The product construction for this specific automaton is rather straightforward. Set aside a copy of your original automaton $A$. (You might add label "many" to each of the states, but that is extra.) Now add to these fresh copies of the original initial state and all states reachable in one step from the initial state. If the initial state has a loop, it is copied twice. (If you fancy it these get extra labels "zero" and "one"). Now remove final states from the "one"s and copy the original wiring.

Like I said, this is just a particular case of the product construction. In fact it might be even what is in your own pseudo code.