If you're wanting to tweak how Simplify works, you might look into TransformationFunctions, which is an option to Simplify.

If you're asking if there is any other function you could use to produce the exact output you want, then I don't know, it's certainly possible. You could also write one yourself, maybe something like:

simple[exp : 
   Times[sum : Plus[__], 
    Power[Plus[1, Times[Power[sum : Plus[__], -2], Power[a_, 2]]], 
     Rational[1, 2]]]] := Sqrt[sum^2 + a^2]

But that looks kinda dicey to me.

But my biggest question is "what is your objective"? You already have an understanding of the expressions you're dealing with, so why are you even bothering with Simplify? If it's for output, then just write your own string-ify-ing function.

Also, The idea of simple can be subjective. How does Mathematica know what you consider simple? The fact that you want two different outputs sort of makes the point here. For each usage, you're going to need to decide what your preferred simplification is. But if you already know how the simplification should look, then why not just start with the expression you want?