Just about anything which will increase friction (bridge to head) will work. I would try something like artists rubber cement on the bottom of the bridge... not the head. The rubber cement will give you the friction to prevent slipping while not affecting the plastic head... But make sure the rubber cement is dry before putting the bridge back on the banjo... The solvent for rubber cement is pretty powerful and may affect the plastic head if still wet. And by the way rubber cement dries pretty fast so drying time is not a problem. I believe contact cement (also an art supply) will do the same thing in spades. Use these on only one surface (the bridge) but not on the head. Instructions will tell you to coat both surfaces but you don't need or want to do that.
I find it odd that you have a problem if you are using steel strings on your banjo.... My bridge has no tendency to slip and I am using a shiny mylar head. On the other hand if you are using nylon strings i could see where you might have a problem as they are much less taught and tend not offer enough pressure on the bridge to prevent slipping. Anything which prevents movement should work as long as it has no effect on the plastic head.
Just what makes the bridge slip? Are you anchoring your pinkie on the side of the bridge? That's a bad habit you should try to break. Otherwise, I agree with everyone else about the downward pressure on the bridge from the tailpiece. Even when I had that bad habit, I rarely moved the bridge.
A mute adds mass. This certainly can cause a bridge to slip if there isn't sufficient downward pressure. Since pressure on the bridge has an effect on tone, a tiny drop of something under the middle foot will do the trick. Pete Seeger uses white glue. Rubber cement will work. I like the idea of rosin but it might not be enough.
It would seem to me that the head material should be considered. A mylar head with the shiney side up might allow the bridge to slip while the coated side would give the bridge a little friction to grip better. An actual skin head would allow little such movement of the bridge....
In fact John Hankley does not give us enough information for an educated guess as to the problem and solution....
It could be that the banjo is tuned so low that there is not enough pressure on the bridge.
The dowel stick may be bent and won't allow proper pressure on the head from the bridge.
If John is using nylon or gut strings instead of steel very little bridge pressure would be exerted and that could enmcourage bridge slippage.
If a minstrel banjo string pressure on the bridge is even lighter. and will often allow slippage of the bridge...In this case it is more a matter of right hand technique and keeping your hand and wrist away from the bridge.
My tnedency is to rest some part of my hand or wrist on the bridge which is not helpful and the problems I have with bridges is usually caused by my bad right hand technique.
The guys at banjo.com told me that if my bridge slipped (and it did), just to lightly sand the feet of the bridge, and then very very lightly sand the head a little, to give it a little tooth and friction.
That cured my own slippage on a Fiberskyn head. When I replaced the head with a Renaissance, I didn't have to perform the sanding ritual. I think head surface is a big factor in the stability of a bridge's location.
I am using steel strings with a Remo Mylar head which is glassy, almost mirror surface, and I have had no problems with bridge slippage. My bridge is a standard wood bridge with three feet and I have had no problems. That is why I am guessing it may be his picking hand placement which may be working the bridge out of position. I shouldn't think you would want to, or have to, roughen the surface of the head to prevent slippage.
If the problem is with a mute,,, Why not pick softer?