You won't get complete sound isolation/sound proofing without building some serious mass. By the time you do that in the closet, there won't be anything left to step into. If the main room is already quiet it may help a little with isolation but only a little.
If you forget sound proofing for the moment, then the idea of the booth is for treatment...cutting down on unwanted echo/reverb and general acoustic improvement. On the upside, the booth is easier/cheaper to handle since it's much smaller. If it were me, treating the entire room to some degree would leave you with a more flexible situation. A third idea I personally like would be to gut the closet, install acoustic foam and a bass trap and building a couple 4'x8' moveable panels. Put the mic in the closet, place the musician in facing the closet, place the moveable panels behing the singer and go for it.
The booth you're working up will at best allow for a stand-up vocalist...put a guitar in there, you're already looking for another option. Where were you planning to mix and monitor? To get that job done as best possible, you'll need a proper listening area as well.
Not busting your chops here. Just throwing out a couple options or ideas.