I bought a Slimline pitts muffler for my ZDZ 50 NG. I noticed there was a bracket to attach the right side (or side opposite of the exhaust manifold) of the muffler to the engine but it requires that I remove one and the head screws and of course replace it to hold the head on and the bracket. Should I use this bracket or is it unnecessary?

The bracket is supplied to help support the muffler and prevent cracks forming in the thin aluminum castings of the muffler. I use the bracket and have found it to be useful. It may not be absolutely necessary, but provides some added insurance against vibration induced cracks in the muffler body.
I have had cracks form in unsupported miufflers, but none so far in the strap supported mufflers.

Use the bracket. There have been quite a few that have cracked or broken because they were unsupported. That's why Slimline put it there.