Unless Browning completely changed the design, the SA-22 stock attaches to the receiver with a nut on the end of the magazine tube, under the buttplate. My SA-22 has no screw behind the grip as you show in the picture.
However, I found another picture of an old SA-22 that shows a hole behind the grip, so I believe Browning must have changed the design.
You might try tightening the nut on the end of the magazine tube. Be careful to keep the magazine tube from turning, or the port won't be aligned with the hole in the side of the stock.
I'm not a Browning historian, but I'm going to guess they decided the second screw wasn't necessary and deleted it in later models.
Edit -
If there was an "old model" and a "new model" stock design, the magazine tube may not tighten the stock like it does with the new models. If that is the case, the only thing I can suggest is to install a washer under the screw head, forcing it to bear on the stock before the screw bottoms out in the receiver.
I think this would be preferable to shortening the threaded end of the screw.
You might have to drill the hole out a little oversized so the washer will fit. A forstner bit might be better than a conventional twist drill bit.