If it's anything like my Gossen Multisix, inside that meter is simply a push to make contact across the flash sync socket. It is in no way connected to the rest of the meter, it just happens to be built into the side of the case, giving the impression that the meter is aware you've pressed it - it isn't.
If it's like that, your meter should work fine if you press to meter, while tripping the flash through some other means (eg, the zip handle of a Lastolite bag, which is designed to poke into a PC plug to trip flash with).
On the other hand, it should be a dead simple fix if you or another were to get inside the meter, if it's anything like mine - just a contact, that's all.