It sounds like there is a cold solder connection in the remote, or there is a crack in the circuit board, thus making a trace opened. It is also possible that a component inside is mechanically damaged.
You can open the remote, and see if there is anything you can find. Pay particular attention to the soldering of the battery contacts. In some remotes, the contacts are mechanically sitting on the board, and are not soldered. In this case, the surfaces must be clean.
The remote you have was not intended to be serviceable. You will have to make sure you remove all the screws to open it. There may be one hidden under the label. Along the seam where the case-shells fit together, there are some interlocking taps. You will have to use a fine tool to release these carefully without damaging the case.
If the remote is not repairable, you can order one from Sony parts. A universal one will not do all the menu functions for your Sony set.