For some reason, Project Mew just sound too reminiscent to Project Mewtwo. In pokemon the first movie, those scientist were on an expedition to find mew in order to study it, and then once discovering some DNA of the creature they used it to clone a second species; Mewtwo.
The fact that the purpose of the Project Mew expedition is to catch and study Mew in a lab tells me that no one will be getting it. I think Mew MUCH too intelligent to allow itself to be the study of mere scientist.
If Goh truly wants to earn the capture then I think he should work his way up to catching lots more pokemon and building a better understanding of how they live and connect with people, they way Ash did through his travels across the regions of the Pokemon World, before becoming a Monarch himself.
The thought of Goh catching Mew and then having it confined to a lab for research just make me think of Mewtwo. In contrast to the Project Mew research, I believe this is why professor Oak's lab is far different than others, as his research facility allows the pokemon to roam and do as they please while he watches from afar to study them.