The MD12 will not drain the battery on any camera it fits unless the button on the handle is constantly pushed dow or the contacts for the remote-control which are located at the lower part of the handle are shortcut by something. The latter happend to me with an FE2. I wouldn't worry to much about it as long as it doesn't happen again. Switch the MD12 to off as you put it away.

If I understand this correctly, the camera fires the shutter and it is just the lightmeter which doesn't work, right?

If you don't have money for a new battery learn to guess the exposure following the tables inside the film box. It is easier than you may think. You were clever enough to buy a camera which does not need batteries to take great photographs.