Adding a return vent to a hot office
We’ve had several similar problems. We’ve solved them by 1) undercut the door, 2) through wall grille,3) through wall grille with sound baffles, 

You’re right, you can’t “push” more air into the room without providing an air relief. (It’s the old straw example...you can’t blow into it if you have your finger over the other end. ) 

1) The simplest and quickest is to cut about 3/4” off the bottom of the door. We’ve done this several times and it works well, except you’ll loose a lot of your sound control. (The 3/4” will about the area of your grille.)

2) providing a through-the-wall grille above the door is also simple and you will also loose some sound control. This option has never turned out good for us. The grille is up near your ears so passerby’s can hear everything better. 

3) We added a grille above a company’s presidents office door with sound baffles and it worked great. We cut the grille in just above the door header on the office side, then extended a duct with sound baffles up the stud space to a grille on the hall side near the ceiling. 

The studs were 16” on center and we used the whole stud space. The duct was about 20” long and had 4 baffles. The baffles are made of strips of duct board mounted perpendicular in the stud space and about 4” apart alternating from side to side. This makes the air travel back and forth through the space. (We often use this design on large supply ducts to trap the sound of the motor noise coming on and going off on the supply side of heating/ac systems.)

It sounds like you have enough air when the doors are open...it’s just a problem when the doors are closed. (The straw example.) So, providing a “relief” vent seems like it will solve your problem. You’ll be “pushing” air out so it should not affect the house return air grille located close by.