Yes, you are right, the air is going to find the low pressure area which should be the pit in the floor. It's the air's path and the velocity getting there that is giving you the problems.
You said that the fan is handling approx. 4000 cfm. If that close to true, your 12" inlet's velocity is over 5100 feet per min. That is very high. By the time it gets to the room because of system losses it will be considerably lower. It also takes a lot of horsepower to do this.
If your object is to put the booth under a positive pressure that doesn't take a lot of air, it takes pressure. I doubt if you would need to go over .5" water column (1psi= 27.68" water col.)
Have you looked at the outlet to see what level of air volume you have there?
If the object is to ventilate the booth, I don't think you are going to like to go the positive route.
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